Notes on Abstract Algebra

Adam Coffman December 15, 2013

Contents

1 Functions 1 2 Binary operations 3 3 Ideals in a Rng 8 4 Rings 12 5 Very elementary algebraic geometry 14

These Notes are compiled from classroom handouts for Math 453 and 553 at IPFW. They are not self-contained, but supplement the required texts, [D], [BB]/[BB3], respectively. Some related books are [H] and [CLO].

1 Functions

Definition 1.1. Given sets S and T, suppose there is a subset G ⊆ S × T with the following properties:

• If (s_{1}, t_{1}) ∈ G and (s_{2}, t_{2}) ∈ G and s_{1} = s_{2} then t_{1} = t_{2};

• For each s ∈ S, there is an element (s, t) ∈ G. Then for each s ∈ S, there is exactly one element α(s) ∈ T so that (s, α(s)) ∈ G. This defines a function α, with domain S and target T, which can be denoted α : S → T. Theorem 1.2. Given S ≠ Ø, and a function α : S → T, the following are equivalent:

1. For all s_{1}, s_{2} ∈ S, if s_{1} ≠ s_{2}, then α(s_{1}) ≠ α(s_{2}) (α has the \underline{one-to-one} property);

2. For any set C and any functions γ : C → S, δ : C → S, if α∘γ : C → T and α∘δ : C → T are the same function, then γ = δ (α has the \underline{left cancellable} property);

3. There is a function β : T → S so that β ∘ α : S → S is equal to the identity function ι : S → S (α has a \underline{left inverse}).

Proof. (3) ⟹ (2): Given α, we assume there is a left inverse β as in (3). If γ and δ satisfy α ∘ γ = α ∘ δ, then β ∘ (α ∘ γ) = β ∘ (α ∘ δ), so by the associative property of composition, (β ∘ α) ∘ γ = (β ∘ α) ∘ δ. It follows that ι ∘ γ = ι ∘ δ, and we get the conclusion γ = δ, so (2) holds.

(2) ⟹ (1): We give a proof of the contrapositive, assuming (1) is false, and then showing (2) is also false. Suppose there are some s_{1}, s_{2} ∈ S so that s_{1} ≠ s_{2} but α(s_{1}) = α(s_{2}). Let C be the two element set {0, 1}, and define the constant function γ : C → S by the graph {(0, s_{1}), (1, s_{1}))} and a different constant function δ : C → S by the graph {(0, s_{2}), (1, s_{2})}. Then for any x ∈ C, (α ∘ γ)(x) = α(γ(x)) = α(s_{1}), and (α ∘ δ)(x) = α(δ(x)) = α(s_{2}). So, α ∘ γ and α ∘ δ are both equal to the constant function C → T with value α(s_{1}) = α(s_{2}). This means (2) is false, since α ∘ γ = α ∘ δ but γ ≠ δ.

(1) ⟹ (3): Assuming (1) holds, consider the graph of α, the subset G ⊆ S × T. The one-to-one property (1) can be re-phrased: if (s_{1}, t_{1}) and (s_{2}, t_{2}) are in G and t_{1} = t_{2} then s_{1} = s_{2}.

We want to construct a function β : T → S. First pick any s_{0} ∈ S (possible since S is non-empty). For any function α with graph G, each t ∈ T falls into one of two possibile types: given t, either there is at least one s ∈ S so that (s, t) ∈ G (t is type I), or there is no s ∈ S so that (s, t) ∈ G (t is type II). Define H ⊆ T × S as the union of the two sets:

H = {(t, s) : t is type I, and (s, t) ∈ G} ∪ {(t, s_{0}) : t is type II}.

For each t ∈ T, there is some point (t, s) ∈ H. To show H is the graph of a function β, consider (t_{1}, s_{1}) ∈ H and (t_{2}, s_{2}) ∈ H; we want to show that if t_{1} = t_{2}, then s_{1} = s_{2}. If t_{1} is type I, then (s_{1}, t_{1}) ∈ G, and if t_{2} = t_{1}, then t_{2} is also type I (since, for example, s = s_{1} satisfies (s, t_{2}) = (s_{1}, t_{1}) ∈ G), so (s_{2}, t_{2}) ∈ G. By the one-to-one property of G, (s_{1}, t_{1}), (s_{2}, t_{2}) ∈ G and t_{1} = t_{2} implies s_{1} = s_{2}. If t_{1} is type II, then (t_{1}, s_{1}) = (t_{1}, s_{0}), and if t_{2} = t_{1}, then t_{2} is also type II (since any s ∈ S satisfying (s, t_{2}) ∈ G would also satisfy (s, t_{1}) ∈ G, and there is no such s), so (t_{2}, s_{2}) = (t_{2}, s_{0}), and s_{2} = s_{0} = s_{1}. We can conclude that the formula: β(t) = s, if t is type I with α(s) = t, and β(t) = s_{0}, if t is type II, defines a function β : T → S.

Considering the composite β ∘ α, pick any s ∈ S, so that (s, α(s)) ∈ G. Then, let t = α(s); since (s, t) ∈ G, t is type I and β(t) is, by definition, an element of S such that (β(t), t) ∈ G. By the one-to-one property of G, (s, α(s)), (β(t), t) ∈ G and α(s) = t implies s = β(t). The conclusion is that s = β(α(s)), so β ∘ α = ι.

Theorem 1.3. Given a function α : S → T, the following are equivalent:

1. For all t ∈ T, there is some s ∈ S so that α(s) = t (α has the \underline{onto} property);

2. For any set C and any functions γ : T → C, δ : T → C, if γ ∘α : S → C and δ ∘α : S → C are the same function, then γ = δ (α has the \underline{right cancellable} property);

3. There is a function β : T → S so that α ∘ β : T → T is equal to the identity function ι : T → T (α has a \underline{right inverse}).

Proof. (3) ⟹ (2): Given α, we assume there is a right inverse β as in (3). If γ and δ satisfy γ ∘ α = δ ∘ α, then (γ ∘ α) ∘ β = (δ ∘ α) ∘ β, so by the associative property of composition, γ ∘ (α ∘ β) = δ ∘ (α ∘ β). It follows that γ ∘ ι = δ ∘ ι, and we get the conclusion γ = δ, so (2) holds.

(2) ⟹ (1): We give a proof of the contrapositive, assuming (1) is false, and then showing (2) is also false. Suppose there is some t_{0} ∈ T so that α(s) ≠ t_{0} for all s ∈ S. Let C be the two element set {0, 1}, and define the constant function γ : T → C by γ(t) = 0 for all t ∈ T, and define a non-constant function δ : T → C, by δ(t) = 0 for all t ≠ t_{0}, and δ(t_{0}) = 1. Then for any s ∈ S, (γ ∘ α)(s) = γ(α(s)) = 0, and (δ ∘ α)(s) = δ(α(s)) = 0, since α(s) ≠ t_{0}. So, γ ∘ α and δ ∘ α are both equal to the constant function S → C with value 0. This means (2) is false, since γ ∘ α = δ ∘ α but γ ≠ δ.

(1) ⟹ (3): Assuming (1) holds, consider the graph of α, the subset G ⊆ S × T. We want to construct a function β : T → S. For each t ∈ T, there is, by the onto property, at least one s ∈ S such that (s, t) ∈ G, so define β(t) ∈ S by choosing any such s. (**)

Considering the composite α ∘ β, for any t ∈ T, (α ∘ β)(t) = α(β(t)) = α(s), where β(t) = s was chosen so that (s, t) ∈ G. This means t = α(s), and the conclusion is that t = α(β(t)), so α ∘ β = ι.

(** footnote remark on this step) If T is an infinite set, this step in the proof requires that we make infinitely many choices, one choice for each t ∈ T. We will assume that we don’t have any problem with doing that, but in some branches of mathematics, the ability to make an infinite sequence of choices isn’t automatically assumed and needs to be taken as another hypothesis for the Theorem.

2 Binary operations

Definition 2.1. Given a set S, a \underline{binary operation on S} is any function from S × S to S. The notation (S, ∗) denotes a set S, together with ∗, a binary operation on S. For x, y ∈ S, and a binary operation ∗, the element ∗((x, y)) will be abbreviated x ∗ y.

Definition 2.2. A binary operation ∗ on S is \underline{associative} means: for all x, y, z ∈ S, (x ∗ y) ∗ z = x ∗ (y ∗ z). It is \underline{commutative} means: for all x, y ∈ S, x ∗ y = y ∗ x.

Definition 2.3. Given (S, ∗), any element e ∈ S such that e ∗ x = x ∗ e = x for all x ∈ S is called an \underline{identity element}.

Proposition 2.4. Given (S, ∗), suppose there is an identity element e ∈ S. Then, the identity element is unique.

Exercise 2.5. Given (S, ∗), with an identity element e, if for all x, y, z ∈ S, x∗(y∗z) = (x∗z)∗y, then ∗ is commutative and associative.

The following problems show that the hypothesis e ∈ S is necessary.

Exercise 2.6. Let S be the set of matrices,

S = \{( \begin{matrix}0 & x & y \\ 0 & 0 & z \\ 0 & 0 & 0 \end{matrix} ) : x, y, z ∈ ℝ\} ,

and let ∗ be matrix multiplication. Show that the formula A ∗ (B ∗ C) = (A ∗ C) ∗ B holds for all matrices A, B, C ∈ S. Show, by giving an explicit numerical example, that ∗ is not commutative on S.

Exercise 2.7. Using the same set S as the previous Exercise, replace the operation ∗ by ⊙, where A⊙B is defined by A∗B −B ∗A. Is ⊙ associative? Does the formula A⊙(B ⊙C) = (A⊙C)⊙B hold for all A, B, C? Is ⊙ commutative?

Exercise 2.8. Give an example of a set and an operation ∗ where x ∗ (y ∗ z) = (x ∗ z) ∗ y holds but ∗ is not associative.

Definition 2.9. Given (S, ∗), and an identity element e ∈ S, and x, y ∈ S, \underline{y is a ∗-inverse for x} means that x ∗ y = y ∗ x = e.

Note that ∗-inverse cannot be defined without an identity element, so in any statement asserting the existence of a ∗-inverse, it is assumed that there exists an identity element for the operation ∗.

Exercise 2.10. Given (S, ∗), let e be an identity element. Then e has a ∗-inverse, and this inverse is unique.

Exercise 2.11. Given (S, ∗), and x ∈ S, if ∗ is associative, and there exists a ∗-inverse for x, then that ∗-inverse for x is unique.

Notation 2.12. Usually it is more convenient to call a ∗-inverse just an “inverse,” and if an element x has a unique inverse, it can be denoted x^{−1}. There may be some other abbreviations for certain operations; customarily a +-inverse of x is denoted −x.

Exercise 2.13. Given (S, ∗), and x, y ∈ S, if ∗ is associative, and x and y both have ∗-inverses, then x ∗ y has a unique ∗-inverse, y^{−1} ∗ x^{−1}.

If ∗ is not associative, (S, ∗) could still have an identity element, but some elements could have more than one inverse.

Example 2.14. Let S = {e, a, b, c}, and define a binary operation ∗ by the table

\begin{matrix}∗ & e & a & b & c \\ e & e & a & b & c \\ a & a & a & e & e \\ b & b & e & a & e \\ c & c & a & e & b \end{matrix}

Note that e is an identity element, and ∗ is not commutative (a ∗ c = e, c ∗ a = a). It is also not associative (a ∗ (a ∗ b) = a ∗ e = a, (a ∗ a) ∗ b = a ∗ b = e). The equations x ∗ b = e, b ∗ x = e have two solutions (x = a and x = c). So, b has an inverse, but b does not have a unique inverse. Since c ∗ b = e, and there is only one solution of the equation c ∗ x = e, c has a unique inverse, c^{−1} = b. Even though a ∗ c = e, a is not an inverse for c because c ∗ a ≠ e.

Exercise 2.15. Given (S, ∗), and x ∈ S, if ∗ is associative, and there exists a ∗-inverse for x, and x ∗ x = x, then x = e.

The above condition that ∗ is associative cannot just be dropped — the conclusion may not follow if ∗ is not associative. In Example 2.14, a has a unique inverse (a ∗ b = b ∗ a = e), but a ∗ a = a, and a ≠ e.

Notation 2.16. Given (S, ∗), x ∈ S, and a natural number n ∈ ℕ, define x^{n} to be x if n = 1, and then for n > 1, recursively define x^{n} to be (x^{(n−1)}) ∗ x. If (S, ∗) has an identity element e ∈ S, then x^{0} is defined to be e. If the element x ∈ S has a unique ∗-inverse x^{−1} ∈ S, and n is a negative integer, then x^{n} is defined to be to (x^{−1})^{(−n)}.

This notation is consistent with the appearance of −1 ∈ ℤ as an exponent in Notation 2.12. The “order of operations” convention should be familiar: x ∗ y^{n} means x ∗ (y^{n}), so that the exponentiation is done before the ∗ operation, and x ∗ y^{n} is, in general, not the same as (x ∗ y)^{n}, where the parentheses indicate the ∗ is done first.

Again, the notation can change for certain operations; when the operation is +, the notation is customarily written as nx, for example, 4x = ((x + x) + x) + x.

Theorem 2.17. Given (S, ∗), with an identity element e, if n is any integer, then e^{n} = e.

Proof. For n = 0 and n = 1, the claim is obvious. Suppose, inductively, that n > 1, and that the claim is true for n − 1, which means e^{n−1} = e. Then e^{n} = e^{n−1} ∗ e, by definition, and this equals e ∗ e by the inductive hypothesis. Then, by the definition of identity element, e ∗ e = e, so e^{n} = e.

Since e is its own unique inverse, negative powers of e are well-defined, and if n is a negative integer, then e^{n} = (e^{−1})^{(−n)} = e^{(−n)} = e.

Exercise 2.18. Given (S, ∗), suppose e is an identity element. If x and y are any elements of S, then (x ∗ y)^{0} = x^{0} ∗ y^{0}, and if n is any nonnegative integer, then (x^{0}) ∗ (x^{n}) = (x^{n}) ∗ (x^{0}) = x^{n}, and e = x^{0} = (x^{0})^{n} = (x^{n})^{0}. If x has a unique inverse, then (x^{0}) ∗ (x^{n}) = (x^{n}) ∗ (x^{0}) = x^{n} and e = x^{0} = (x^{0})^{n} = (x^{n})^{0} for all n ∈ ℤ.

Hint. These are easy to prove — but it is worth remarking that these statements, and Theorem 2.17, do not require ∗ to be associative.

Lemma 2.19. Given (S, ∗), and x, y ∈ S, if ∗ is associative and x∗y = y ∗x then x∗y^{n} = y^{n} ∗x for all n ∈ ℕ. If, also, y has a ∗-inverse, then x ∗ y^{n} = y^{n} ∗ x for all n ∈ ℤ.

Proof. Note that ∗ doesn’t have to be a commutative operation: we’re only assuming that x and y commute with each other and not necessarily with other elements of S.

The claim is clearly true for n = 1 (and does not even require ∗ to be associative).

For n > 1, assume that the statement has been proved for n − 1: the “inductive hypothesis” is that x ∗ y^{n−1} = y^{n−1} ∗ x, and we want to show that x ∗ y^{n} = y^{n} ∗ x.

Using the definition of nth power and the associative law, x∗y^{n} = x∗(y^{n−1}∗y) = (x∗y^{n−1})∗y, and the inductive hypothesis says this equals (y^{n−1} ∗ x) ∗ y, which, using the associativity and the x ∗ y = y ∗ x hypothesis, equals y^{n−1} ∗ (x ∗ y) = y^{n−1} ∗ (y ∗ x) = (y^{n−1} ∗ y) ∗ x, which by definition is y^{n} ∗ x.

If e ∈ S is an identity element, then x∗y^{0} = y^{0}∗x, even if ∗ is not associative and x∗y ≠ y∗x. If y has an inverse, it is unique by Exercise 2.11, and the n = −1 claim is that if x ∗ y = y ∗ x,

then x ∗ y^{−1} = y^{−1} ∗ x, which is proved by the following steps:

x ∗ y^{−1} = (e ∗ x) ∗ y^{−1} = ((y^{−1} ∗ y) ∗ x) ∗ y^{−1} = (y^{−1} ∗ (y ∗ x)) ∗ y^{−1}
= (y^{−1} ∗ (x ∗ y)) ∗ y^{−1} = y^{−1} ∗ ((x ∗ y) ∗ y^{−1}) = y^{−1} ∗ (x ∗ (y ∗ y^{−1}))
= y^{−1} ∗ (x ∗ e) = y^{−1} ∗ x.

If n is any negative integer, then x∗y^{n} = x∗(y^{−1})^{−n} by definition, and since x∗y^{−1} = y^{−1} ∗x (which was just proved), the first part of the Theorem applies to x, y^{−1}, and the positive integer −n, giving x ∗ (y^{−1})^{−n} = (y^{−1})^{−n} ∗ x, so x ∗ y^{n} = y^{n} ∗ x.

The associativity is a necessary hypothesis. In Example 2.14, b∗c = c∗b, but c∗b^{2} = c∗a = a, and b^{2} ∗ c = a ∗ c = e.

Theorem 2.20. Given (S, ∗), and x, y ∈ S, if ∗ is associative and x ∗ y = y ∗ x then (x ∗ y)^{n} = x^{n} ∗ y^{n} for all n ∈ ℕ. If, also, x and y have ∗-inverses, then (x ∗ y)^{n} = x^{n} ∗ y^{n} holds for all n ∈ ℤ.

Proof. The statement is clearly true for n = 1 (and does not even require associativity or x ∗ y = y ∗ x).

For n > 1, assume that the statement has been proved for n − 1: the “inductive hypothesis” is that (x ∗ y)^{n−1} = x^{n−1} ∗ y^{n−1}, and we want to show that (x ∗ y)^{n} = x^{n} ∗ y^{n}. By definition, (x∗y)^{n} = ((x∗y)^{n−1})∗(x∗y), and this equals (x^{n−1} ∗y^{n−1})∗(x∗y) by the inductive hypothesis. Then, by associativity, it equals ((x^{n−1} ∗ y^{n−1}) ∗ x) ∗ y = (x^{n−1} ∗ (y^{n−1} ∗ x)) ∗ y. Here, Lemma 2.19 applies, which is where the x ∗ y = y ∗ x hypothesis is needed, to give (x^{n−1} ∗ (x ∗ y^{n−1})) ∗ y, which by associativity equals ((x^{n−1} ∗ x) ∗ y^{n−1}) ∗ y = (x^{n−1} ∗ x) ∗ (y^{n−1} ∗ y), and by definition, equals x^{n}y^{n}.

The second claim allows n to be 0; since x and y have inverses, there must be an identity element e, and then Exercise 2.18 applies, even if x and y don’t commute. To prove the second claim for negative n, x, y, and x ∗ y have unique inverses, by Exercises 2.11 and 2.13 (which also require associativity), and (x ∗ y)^{n} is defined to be ((x ∗ y)^{−1})^{(−n)}. By the x ∗ y = y ∗ x hypothesis and Exercise 2.13, this is equal to ((y ∗ x)^{−1})^{(−n)} = (x^{−1} ∗ y^{−1})^{(−n)}, which by the first claim of the Theorem, is equal to (x^{−1})^{(−n)} ∗ (y^{−1})^{(−n)} = x^{n} ∗ y^{n}.

Corollary 2.21. Given (S, ∗), and x, y ∈ S, if ∗ is associative and y is a ∗-inverse of x, then y^{n} is a ∗-inverse of x^{n} and x^{(−n)} = y^{n} for all n ∈ ℤ.

Proof. By the uniqueness of inverses for associative operations (Exercise 2.11), y is the unique inverse of x (y = x^{−1}) and x is the unique inverse of y (x = y^{−1}). Since x and y have inverses and commute by definition (x ∗ y = y ∗ x = e), Theorem 2.20 applies to give x^{n} ∗ y^{n} = (x ∗ y)^{n} = e^{n}, and e^{n} = e by Theorem 2.17. Similarly, y^{n} ∗ x^{n} = (y ∗ x)^{n} = e^{n} = e. It can be concluded that y^{n} = (x^{n})^{−1} = (x^{−1})^{n}. The proof of the second equality needs a few cases: the n = 0 case is easy, and if n > 0, then x^{(−n)} is defined to be (x^{−1})^{−(−n)} = y^{n}. If n < 0, then y^{n} is defined to be (y^{−1})^{(−n)} = x^{(−n)}.

The associativity is a necessary hypothesis for both the Theorem and the Corollary. In Example 2.14, b ∗ a = a ∗ b = e, but b^{2} ∗ a^{2} = a ∗ a = a ≠ e = (b ∗ a)^{2}.

Theorem 2.22. Given (S, ∗), and g ∈ S, if ∗ is associative, then (g^{a}) ∗ (g^{b}) = g^{(a+b)} for any a, b ∈ ℕ. If g has a ∗-inverse, then (g^{a}) ∗ (g^{b}) = g^{(a+b)} for any a, b ∈ ℤ.

Proof. The claim is true for any a if b = 1, since (g^{a})∗(g^{1}) = (g^{a})∗g = g^{(a+1)} by definition of nth power for n = a + 1. Suppose, inductively, that b > 1, and that the claim is true for b − 1, which means (g^{a})∗(g^{(b−1)}) = g^{(a+(b−1))}. Then (g^{a})∗(g^{b}) = (g^{a})∗((g^{(b−1)})∗g) by definition of bth power, which equals ((g^{a}) ∗ (g^{(b−1)})) ∗ g by associativity, and equals (g^{(a+(b−1))}) ∗ g by the inductive hypothesis. Then the definition of nth power again shows this is equal to g^{(a+(b−1))+1} = g^{(a+b)}, which proves the claim for the positive integer b.

If e ∈ S is an identity element, the claim for a = 0 or b = 0 was stated in Exercise 2.18. If g has an inverse, it is unique by Exercise 2.11. The rest of the proof proceeds in several

cases. Case 1. If a and b are both negative, then g^{a} ∗ g^{b} = ((g^{−1})^{−a}) ∗ ((g^{−1})^{−b}), and the first claim applies to the positive exponents −a, −b, and g^{−1} ∈ S, so ((g^{−1})^{−a}) ∗ ((g^{−1})^{−b}) = (g^{−1})^{(−a)+(−b)} = g^{−((−a)+(−b))} = g^{a+b}.

There are several remaining cases, when a and b have opposite signs. Case 2. 0 < −a = b. g^{a} ∗ g^{b} = ((g^{−1})^{−a}) ∗ g^{b} = ((g^{−1})^{b}) ∗ g^{b} = e, by Cor. 2.21. Then, e = g^{0} = g^{a+b} by definition of g^{0}. Case 3. 0 < a = −b. g^{a}∗g^{b} = g^{a}∗((g^{−1})^{−b}) = g^{a}∗((g^{−1})^{a}) = e = g^{a+b}. Case 4. 0 < −a < b. g^{a} ∗ g^{b} = ((g^{−1})^{−a}) ∗ ((g^{−a}) ∗ (g^{a+b})), using the claim for the positive exponents −a, b + a, and b = (−a) + (b + a). Then, by associativity, the product is equal to (((g^{−1})^{−a}) ∗ (g^{−a})) ∗ (g^{a+b}) = e ∗ (g^{a+b}) = g^{a+b}. Case 5. 0 < b < −a. g^{a} ∗ g^{b} = ((g^{−1})^{−a}) ∗ g^{b} = (((g^{−1})^{−a−b}) ∗ ((g^{−1})^{b})) ∗ g^{b} = ((g^{−1})^{−(a+b)})∗(((g^{−1})^{b})∗g^{b}) = (g^{a+b})∗e = g^{a+b}. Case 6. 0 < a < −b. g^{a} ∗g^{b} = g^{a} ∗((g^{−1})^{−b}) = g^{a} ∗ (((g^{−1})^{a}) ∗ ((g^{−1})^{(−a)+(−b)})) = (g^{a} ∗ ((g^{−1})^{a})) ∗ ((g^{−1})^{−(a+b)}) = e ∗ (g^{a+b}) = g^{a+b}. Case 7. 0 < −b < a. g^{a} ∗ g^{b} = ((g^{a+b}) ∗ (g^{−b})) ∗ ((g^{−1})^{−b}) = (g^{a+b}) ∗ ((g^{−b}) ∗ ((g^{−1})^{−b})) = (g^{a+b}) ∗ e = g^{a+b}.

Example 2.23. The associativity hypothesis is necessary. Try, for example, S = ℕ, and let ∗ be the operation of integer exponentiation (∗ = ˆ ), so m ∗ n is m multiplied (·) by itself n times, and ∗ is not associative, for example, (2 ∗ 3) ∗ 2 = 64, and 2 ∗ (3 ∗ 2) = 512. Theorem 2.22 doesn’t work, where “3^{3}” is defined to be (3 ∗ 3) ∗ 3 = 27 · 27 · 27 = 19683. This is the same as (3^{2}) ∗ (3^{1}), but not the same as (3^{1}) ∗ (3^{2}) = 3 ∗ 27 = 3 · 3 · 3 ⋯ 3 = 7625597484987.

Theorem 2.24. Given (S, ∗), and g ∈ S, if ∗ is associative, then g^{(mn)} = (g^{m})^{n} for any m, n ∈ ℕ. If g has a ∗-inverse, then g^{(mn)} = (g^{m})^{n} for any m, n ∈ ℤ.

Proof. The claim is true for any m if n = 1, since g^{(m·1)} = g^{m} = (g^{m})^{1}. Suppose, inductively, that n > 1, and that the claim is true for n − 1, which means g^{(m(n−1))} = (g^{m})^{(n−1)}. Then (g^{m})^{n} = ((g^{m})^{(n−1)})∗(g^{m}) by the definition of nth power, which equals (g^{(m(n−1))})∗(g^{m}) by the inductive hypothesis. By Theorem 2.22 on adding exponents, (g^{(m(n−1))})∗(g^{m}) = g^{(m(n−1))+m} = g^{mn}, which proves the claim for the positive integer n.

If there is an identity element e, Exercise 2.18 applies to prove the second claim in the case that m = 0 or n = 0. It remains to check the cases where m or n is negative. If m is any integer, and n is positive, the above proof by induction will work even for negative m as long as g has an inverse, using Theorem 2.22 to add any integer exponents.

If m is positive, and n is negative, then mn is negative, so by definition, g^{(mn)} = (g^{−1})^{−(mn)} = (g^{−1})^{(m(−n))}, and since m and −n are positive, the first claim gives ((g^{−1})^{m})^{(−n)}, and then Corollary 2.21 gives ((g^{m})^{−1})^{(−n)} = (g^{m})^{n}. If m and n are both negative, then g^{(mn)} = g^{(−m)(−n)} = (g^{(−m)})^{(−n)} by the first claim, and then Corollary 2.21 again gives ((g^{m})^{−1})^{(−n)} = (g^{m})^{n}.

The associativity is a necessary hypothesis here, too. In Example 2.14, c^{4} = ((c ∗ c) ∗ c) ∗ c = (b ∗ c) ∗ c = e ∗ c = c, but (c^{2})^{2} = (c ∗ c) ∗ (c ∗ c) = b ∗ b = a.

3 Ideals in a Rng

Definition 3.1. A set S, together with two binary operations +_{S} and ·_{S}, is a \underline{rng} means that +_{S} and ·_{S} are associative, +_{S} is commutative, there is an element 0_{S} which is an identity element for (S, +_{S}), and every x ∈ S has a +_{S}-inverse, denoted −_{S}x, or just −x. The two operations are related by the distributive laws: for all a, b, c ∈ S, a · (b + c) = (a · b) + (a · c) and (a + b) · c = (a · c) + (b · c).

If ·_{S} is commutative, S is a \underline{commutative rng}.

Given x ∈ S, a +_{S}-inverse −x can be called the \underline{opposite} of x in S; it is unique by Exercise 2.11. Subtraction in S is defined by x − y = x −_{S} y = x +_{S} (−y) = x + (−y).

Definition 3.2. Let (R, +_{R}, ·_{R}) be a rng. A set S is a \underline{subrng} of R means:

• S ⊆ R, and

• There are operations +_{S} and ·_{S} such that (S, +_{S}, ·_{S}) is a rng, and

• For all x, y ∈ S, x +_{R} y = x +_{S} y, and

• For all x, y ∈ S, x ·_{R} y = x ·_{S} y.

Theorem 3.3. If S is a subrng of R, where R has zero element 0_{R}, then 0_{R} is an element of S, and is equal to the zero element of S.

Proof. By the second part of Definition 3.2, S is a rng, so by Definition 3.1 applied to S, S contains a zero element 0_{S} ∈ S. By the first part of Definition 3.2, S ⊆ R, which implies 0_{S} ∈ R. By Definition 3.1 applied to S, 0_{S} +_{S} 0_{S} = 0_{S}, and by Definition 3.2, 0_{S} +_{R} 0_{S} = 0_{S} +_{S} 0_{S}. It follows that 0_{S} +_{R} 0_{S} = 0_{S} ∈ R, and then Exercise 2.15, applied to the associative operation +_{R} and the element 0_{S} ∈ R, which has a +_{R}-inverse in R, implies 0_{S} = 0_{R}.

Theorem 3.3 can be used in this way: if S is a set that does \underline{not} contain 0_{R} as one of its elements, then S is \underline{not} a subrng of R.

Theorem 3.4. If S is a subrng of a rng R, then for every w ∈ S, the opposite of w in S is the same as the opposite of w in R.

Proof. Let w be an element of S; then w ∈ R because S ⊆ R.

First, we show that an additive inverse of w in S is also an additive inverse of w in R. Let y be any additive inverse of w in S, meaning y ∈ S and w +_{S} y = 0_{S}. (There exists at least one such y, by Definition 3.1 applied to S.) S ⊆ R implies y ∈ R. From Theorem 3.3, 0_{S} = 0_{R}, and w +_{S} y = w +_{R} y by Definition 3.2, so w +_{R} y = 0_{R}, which means y is an additive inverse of w in R.

Second, we show that an additive inverse of w in R is also an additive inverse of w in S. Let z be any additive inverse of w in R, meaning z ∈ R and w +_{R} z = 0_{R}. (There exists at least one such z, by Definition 3.1 applied to R.) Then w +_{R} z = 0_{R} = w +_{R} y, so by Left Cancellation in R ([BB3] Prop. 3.1.7), z = y and y ∈ S, which imply z ∈ S and w +_{S} z = w +_{S} y = 0_{S}, meaning z is an additive inverse of w in S.

By uniqueness of opposites (Exercise 2.11 applied to either R or S), we can refer to y = z as “the” opposite of w, and denote it y = −w.

Theorem 3.4 also implies that subtraction in S is the same as subtraction in R: by the above definition, for v, w ∈ S, v −_{S} w = v +_{S} y = v +_{R} y = v −_{R} w.

Theorem 3.4 can be used in this way: if S is a subset of a rng R and there is an element w ∈ S, where the opposite of w in R is \underline{not} an element of S, then S is \underline{not} a subrng of R.

Theorem 3.5. Let (R, +_{R}, ·_{R}, 0_{R}) be a rng, and let S be a subset of R. Then S, with the same addition and multiplication operations, is a subrng of R if and only if:

(1) x ∈ S, y ∈ S imply x +_{R} y ∈ S (closure under +_{R} addition), and

(2) x ∈ S implies the opposite of x in R, −_{R}x, is an element of S (closure under −_{R} opposite), and

(3) x ∈ S, y ∈ S imply x ·_{R} y ∈ S (closure under ·_{R} multiplication), and

(4) S ≠Ø.

Proof. Let R have zero element 0_{R}.

First suppose S is a subrng, so that as in the Proof of Theorem 3.3, S contains a zero element 0_{S}, which shows S ≠Ø, and (4) is true. From Definition 3.1, x ∈ S, y ∈ S imply x +_{S} y ∈ S, and from Definition 3.2, x +_{S} y = x +_{R} y, so x +_{R} y ∈ S, establishing (1). From Definition 3.1, x ∈ S implies −_{S}x ∈ S, and by Theorem 3.4, because S is a subrng, −_{S}x = −_{R}x, so −_{R}x ∈ S, establishing (2). Similarly, from Definition 3.1, x ∈ S, y ∈ S implies x ·_{S} y ∈ S, and from the definition of subrng, x ·_{S} y = x ·_{R} y, so x ·_{R} y ∈ S, establishing (3).

Conversely, it follows from (1), (2), (3), and (4) that S is a subrng of R, as follows: R is a rng, and S is a subset of R by hypothesis. Define +_{S} and ·_{S} by x +_{S} y = x +_{R} y, and x ·_{S} y = x ·_{R} y — these define operations on S by (1) and (2) (so S is closed under +_{S} and ·_{S}, which is required for S to be a rng as in Definitions 3.1 and 3.2), but it remains to check the other properties to show that (S, +_{S}, ·_{S}) is a rng. Since S ≠Ø by (4), there is some x ∈ S, and by (2), −_{R}x ∈ S. By (1), x +_{R} (−_{R}x) = 0_{R} ∈ S. 0_{R} ∈ S satisfies x +_{S} 0_{R} = x +_{R} 0_{R} = x for all x ∈ S, so 0_{R} is a zero element for S. It follows that −_{R}x is an additive inverse of x in S: x +_{S} (−_{R}x) = x +_{R} (−_{R}x) = 0_{R} = 0_{S}. The other rng properties, associativity of +_{S} and ·_{S}, commutativity of +_{S}, and the distributive properties, follow immediately from the facts that these properties hold in R and the operations in S give the same sums and products.

Property (2) in Theorem 3.5 is necessary; for example, the set of even integers is a rng, and the subset of positive evens is non-empty and closed under addition and multiplication, but is not a subrng.

Definition 3.6. Given a rng R and a subset I ⊆ R, I is an \underline{ideal subrng} means: I is a subrng, with the property that for any x ∈ I and any r ∈ R, the products x · r and r · x are in I.

Such a set is also called just an \underline{ideal}, or a \underline{two-sided ideal}. If R is a commutative rng, then x · r ∈ I is enough to imply the other condition. However, if R is not commutative, one can define \underline{left ideals} requiring only r · x ∈ I, and \underline{right ideals} where x · r ∈ I.

To tell whether a set I is an ideal of R, it’s enough to check some of the properties, and then the rest follow automatically (as in Theorem 3.5):

Proposition 3.7. Given a rng R and a subset I ⊆ R, I is an ideal of R if and only if (1) I ≠Ø, (2) if x, y ∈ I, then x +_{R} y ∈ I, (3) if a ∈ I, then its additive inverse −_{R}a in R satisfies −_{R}a ∈ I, and (4) if x ∈ I and r ∈ R, then x · r ∈ I and r · x ∈ I.

Notation 3.8. Given a rng R, let S be any subset of R. Define ⟨S⟩ ⊆ R to be the intersection of all ideals I such that S ⊆ I. S is called a \underline{generating set} or a \underline{basis} of ⟨S⟩. By the following Lemma, ⟨S⟩ can be called the ideal \underline{generated by S}.

Lemma 3.9. For any set S, the set ⟨S⟩ is an ideal in R. If I is an ideal and S ⊆ I, then ⟨S⟩ ⊆ I.

Sketch of Proof. It is easy to check that the intersection of two ideals is an ideal, and the proof that the intersection of any collection of ideals is an ideal is a straightforward generalization. The second part of the Lemma follows immediately from the definition of intersection, and it means that ⟨S⟩ is the “smallest” ideal containing S.

Definition 3.10. If I is an ideal in R, and there exists some a ∈ R such that I = ⟨{a}⟩, then I is called a \underline{principal} ideal.

Definition 3.11. If every ideal I in R is a principal ideal, then R is called a \underline{principal ideal rng}.

Definition 3.12. If I is an ideal in R, and there exists some finite set {a_{1}, a_{2}, …, a_{n}} ⊆ R such that I = ⟨\{a_{1}, a_{2}, …, a_{n}\}⟩, then I is called a \underline{finitely generated} ideal.

⟨{a_{1}, a_{2}, …, a_{n}}⟩ is usually abbreviated ⟨a_{1}, a_{2}, …, a_{n}⟩.

Proposition 3.13 ([H], §III.2).

⟨a⟩ = {r · a + a · s + n · a + ∑_{i=1}^{m} r_{i} · a · s_{i} : r, s, r_{i}, s_{i} ∈ R, n ∈ ℤ, m ∈ ℕ}.

As in Notation 2.16, the term n · a refers to a + a + … + a (n terms) if n ∈ ℕ, 0 ∈ R if n = 0, and (−a) + (−a) + … + (−a) (−n terms) if −n ∈ ℕ.

Proposition 3.14. Given a rng R, if a is an element of the center of R (meaning a · r = r · a for all r ∈ R) then

⟨a⟩ = {r · a + n · a : r ∈ R, n ∈ ℤ}.

For example, in a commutative rng R, every a ∈ R is in the center.

Proposition 3.15. Given a rng R and an element a ∈ R, the set Ra = {r · a : r ∈ R} is a left ideal of R.

However, the set Ra need not contain a, for example if R = 2ℤ and a = 2, then a ∉ Ra = 4ℤ.

Exercise 3.16. Given a rng R, let I and J be ideals in R. Show by an example that I ∪ J is not necessarily an ideal in R. Show that I ∪ J ⊆ I + J. Show that ⟨I ∪ J⟩ = I + J.

Theorem 3.17. Given a rng R, let I and J be ideals in R. The following set is an ideal:

{∑_{i=1}^{n} f_{i}g_{i} : f_{i} ∈ I, g_{i} ∈ J, n ∈ ℕ} ,

and it is equal to the ideal ⟨{f · g : f ∈ I, g ∈ J}⟩.

Proof. To show it’s an ideal, use Proposition 3.7. The set contains the zero element: 0 = 0 · 0, a sum with one term (n = 1). The set is closed under sums: for any a_{1}, … a_{k} ∈ I, b_{1}, … b_{k} ∈ J, A_{1}, …, A_{j} ∈ I, B_{1}, … B_{j} ∈ J, define lists α_{i} and β_{i} by α_{i} = a_{i} if 1 ≤ i ≤ k, α_{i} = A_{i−k} if k + 1 ≤ i ≤ k + j, and β_{i} = b_{i} if 1 ≤ i ≤ k, β_{i} = B_{i−k} if k + 1 ≤ i ≤ k + j. Then α_{i} ∈ I, and β_{i} ∈ J, and

∑_{ℓ=1}^{k} a_{ℓ}b_{ℓ} + ∑_{m=1}^{j} A_{m}B_{m} = ∑_{i=1}^{k+j} α_{i}β_{i}.

The set is also closed under additive inverses: − ∑ a_{i}b_{i} = ∑(−a_{i})b_{i}, since −a_{i} ∈ I. Finally, it has the two-sided ideal properties: for any r ∈ R, (∑ a_{i}b_{i}) · r = ∑ a_{i}(b_{i} · r) is in the set, since b_{i} · r ∈ J, and similarly, r · (∑ a_{i}b_{i}) = ∑(r · a_{i})b_{i} is in the set, since r · a_{i} ∈ I.

So, this set is an ideal, and it contains every product of the form f ·g (single-term sums), and Lemma 3.9 says ⟨{f · g}⟩ ⊆ {∑ f_{i}g_{i}}. To show the other subset relation, consider any ∑ f_{i}g_{i}, with f_{i} ∈ I, g_{i} ∈ J. If X is any ideal containing all the products {f · g}, then f_{i} · g_{i} ∈ X, and X is closed under sums, so ∑ f_{i}g_{i} ∈ X. So, any ideal containing all the products {f · g} must contain the sum ∑ f_{i}g_{i}, and the intersection of all of these ideals X must also contain ∑ f_{i}g_{i}, which means ∑ f_{i}g_{i} ∈ ⟨{f · g : f ∈ I, g ∈ J}⟩. This shows the sets are equal.

Definition 3.18. The ideal from the previous Theorem is denoted IJ, the \underline{product} ideal.

Exercise 3.19. Given a rng R, and I, J ideals in R, IJ ⊆ I ∩ J.

Definition 3.20. Given a commutative rng R and any ideal I in R, the \underline{radical} of I is the set

\sqrt{I} = \{f ∈ R : ∃m ∈ ℕ : f^{m} ∈ I\}.

The quantity f^{m} refers to f multiplied by itself m times, as in Notation 2.16. Note I ⊆ \sqrt{I} ⊆ R.

Proposition 3.21. Given a commutative rng R and any ideal I in R, \sqrt{I} is an ideal in R.

Definition 3.22. Given a commutative rng R and any ideal I in R, if I = \sqrt{I} then I is called a \underline{radical} ideal.

Proposition 3.23. Given a commutative rng R and any ideal I in R, \sqrt{\sqrt{I}} = \sqrt{I}, so \sqrt{I} is always a radical ideal.

Exercise 3.24. Given a commutative rng R and any ideals I, J in R, show that \sqrt{I ∩ J} = \sqrt{I} ∩ \sqrt{J}.

Theorem 3.25. Given any set Y, and any rng R, the set of functions M = {f : Y → R} is a rng. If B ⊆ Y is any subset, and K ⊆ M is a subrng, then the set I(B) = {f ∈ K : f(x) = 0_{R} for all x ∈ B} is an ideal in K. If B ⊆ C ⊆ Y, then I(C) ⊆ I(B).

Proof. First, the usual sum and product of functions define operations so that M is a rng. By Theorem 3.5, every subrng K of M contains 0_{M}, which is the constant zero function (x ↦ 0_{R} for all x ∈ Y). The set I(B) always contains 0_{M}. To show I(B) is an ideal, it is easy to check that if f and g are in I(B) then −f and f + g are in I(B), and if r ∈ K, then f · r ∈ I(B).

If f ∈ I(C), then f(x) = 0_{R} for all x ∈ C. In particular, f(x) = 0_{R} for all x ∈ B ⊆ C, so f ∈ I(B). This shows I(C) ⊆ I(B).

4 Rings

Definition 4.1. A set S, together with two binary operations +_{S} and ·_{S}, is a \underline{ring} means that +_{S} and ·_{S} are associative, there is an element 0_{S} ∈ S which is an identity element for (S, +_{S}), there is an element 1_{S} ∈ S which is an identity element for (S, ·_{S}), and every x ∈ S has a +_{S}-inverse, denoted −_{S}x, or just −x. The two operations are related by the distributive laws: for all a, b, c ∈ S, a · (b + c) = (a · b) + (a · c) and (a + b) · c = (a · c) + (b · c).

Theorem 4.2. If S is a ring, then S is a rng.

Proof. The only part of the definition of rng missing from Definition 4.1 is the commutativity of +_{S}. Let 1 be the identity element for ·_{S}, and consider v, w ∈ S. We start with (1+1)·(v+w) ∈ S, set LHS=RHS, and use both distributive laws:

(1 + 1) · (v + w) = (1 + 1) · (v + w)
((1 + 1) · v) + ((1 + 1) · w) = (1 · (v + w)) + (1 · (v + w))
((1 · v) + (1 · v)) + ((1 · w) + (1 · w)) = (v + w) + (v + w)
(v + v) + (w + w) = (v + w) + (v + w).

Then, the associative law gives v + (v + (w + w)) = v + (w + (v + w)), and Left Cancellation leaves v + (w + w) = w + (v + w). Using the associative law again, (v + w) + w = (w + v) + w, and Right Cancellation gives the result v + w = w + v.

So, the addition operation for any ring S defines an abelian group (S, +, 0). If, additionally, ·_{S} is a commutative operation, S is a \underline{commutative ring}.

Some authors use the term “ring” to mean a rng as in Definition 3.1, and then use the above Definition 4.1 to define a “ring with identity.”

Definition 4.3. Let (R, +_{R}, ·_{R}, 0_{R}, 1_{R}) be a ring. A set S is a \underline{subring} of R means:

• S ⊆ R, and

• There are operations +_{S} and ·_{S} such that (S, +_{S}, ·_{S}, 0_{S}, 1_{S}) is a ring, and

• For all x, y ∈ S, x +_{R} y = x +_{S} y, and

• For all x, y ∈ S, x ·_{R} y = x ·_{S} y, and

• 1_{S} = 1_{R}.

Note that it follows from Definition 4.3 that if S is a subring of R, then for x, y ∈ S, the sum and product x +_{R} y and x ·_{R} y are in S. The converse of this statement is false (for example, the positive integers ℕ are closed under sums and products in ℤ, but ℕ isn’t a ring). The following Theorem claims that closure under sums, products, and additive inverses, and containment of R’s multiplicative identity element, is enough for non-empty subsets to be subrings.

Theorem 4.4. Let (R, +_{R}, ·_{R}, 0_{R}, 1_{R}) be a ring, and let S be a subset of R, with the same addition and multiplication operations. The following are equivalent:

• S is a subring of R; • S is a subrng of R and 1_{R} ∈ S;

• (1) x ∈ S, y ∈ S imply x +_{R} y ∈ S (closure under +_{R} addition), and (2) x ∈ S implies the opposite of x in R, −_{R}x, is an element of S (closure under −_{R} opposite), and (3) x ∈ S, y ∈ S imply x ·_{R} y ∈ S (closure under ·_{R} multiplication), and (4) 1_{R} ∈ S.

Proof. Assuming 1_{R} ∈ S, it follows that S is non-empty, and the equivalence of the second and third statements is Theorem 3.5.

Suppose S is a subring of R. Then S is a ring by Definition 4.3, and R and S are rngs by Theorem 4.2, so S is a subrng of the rng R as in Definition 3.2. S contains 1_{S} because S is a ring, 1_{S} = 1_{R} by Definition 4.3, so 1_{R} ∈ S.

Conversely, suppose S is a subrng of R and 1_{R} ∈ S. 0_{S} = 0_{R} as in Theorem 3.5. For all s ∈ S, 1_{R} ·_{S} s = 1_{R} ·_{R} s = s = s ·_{R} 1_{R} = s ·_{S} 1_{R}, so 1_{R} is an identity element for ·_{S}: let 1_{S} = 1_{R}. So, S is a rng with a multiplicative identity: (S, +_{S}, ·_{S}, 0_{R}, 1_{R}) is a ring, and satisfies the definition of subring.

For the rest of this Section, let R be a ring, with operations + and ·, additive identity 0_{R} = 0, multiplicative identity 1_{R} = 1, and additive inverses −x.

Definition 4.5. An \underline{integral domain} is a commutative ring R such that for all a, b ∈ R, ab = 0 ⟹ a = 0 or b = 0.

Definition 4.6. If a ring R is an integral domain and also a principal ideal rng, then R is called a \underline{principal ideal domain}, or a \underline{p.i.d.}.

Proposition 4.7. Given a ring R and an element a ∈ R,

⟨a⟩ = {∑_{i=1}^{m} r_{i} · a · s_{i} : r_{i}, s_{i} ∈ R, m ∈ ℕ}.

Proposition 4.8. Given a ring R, and an element a ∈ R, the set Ra = {r · a : r ∈ R} is a left ideal of R, and a ∈ Ra.

Proof. This follows from Proposition 3.15, and because 1 ∈ R, 1 · a = a ∈ Ra.

Proposition 4.9. Given a ring R and an element a of the center of R, then ⟨a⟩ = Ra = aR.

Proposition 4.10. Given a ring R and any non-empty subset X in the center of R, then

⟨X⟩ = {∑_{i=1}^{n} r_{i} · x_{i} : r_{i} ∈ R, x_{i} ∈ X, n ∈ ℕ}.

Note that X could be an infinite set, but all the elements of ⟨X⟩ are just finite sums. As a trivial case, ⟨ Ø ⟩ = {0_{R}}.

Proposition 4.11. Given a ring R, if I is an ideal of R, and 1 ∈ I, then I = R.

Lemma 4.12. If R is an integral domain, then {0_{R}} is a radical ideal.

Proof. Suppose R is an integral domain. Then, by Definition 3.22, “{0_{R}} is a radical ideal” means that if a^{n} ∈ \{0_{R}\}, then a ∈ \{0_{R}\}. Elements a such that a^{n} = 0_{R} are called \underline{nilpotent}, and we need to show that 0_{R} is the only nilpotent element of R. Statements involving n ∈ ℕ require induction: if n = 1, then a^{1} = 0_{R} ⟹ a = 0_{R}, so the n = 1 case is true. Suppose that the only element x ∈ R such that x^{n} = 0_{R} is x = 0_{R}. If a^{n+1} = 0_{R}, then a · a^{n} = 0_{R}, and by the integral domain property, either a = 0_{R}, or a^{n} = 0_{R}. Either way, a = 0_{R} by the inductive hypothesis.

Theorem 4.13. Given any set Y, and any ring R, the set of functions M = {f : Y → R} is a ring. If B ⊆ Y is any subset, and K ⊆ M is a subring, then the set I(B) = {f ∈ K : f(x) = 0_{R} for all x ∈ B} is an ideal in K. If B ⊆ C ⊆ Y, then I(C) ⊆ I(B). If R is an integral domain, then I(B) is a radical ideal.

Proof. First, the usual sum and product of functions define operations so that M is a ring, as in Theorem 3.25. By Definition 4.3, every subring K of M contains 1_{M}, which is the constant function (x ↦ 1_{R} for all x ∈ Y). The first two claims about the ideal I(B) are exactly as in Theorem 3.25.

To show that I(B) is radical, suppose f ∈ \sqrt{I(B)}, so that f^{m} ∈ I(B). Then, for any x ∈ B, f^{m}(x) = 0_{R} = (f(x))^{m}, and by Lemma 4.12 about integral domains, f(x) = 0_{R}, so f ∈ I(B).

5 Very elementary algebraic geometry

Notation 5.1. Let F be a field with additive identity 0 and multiplicative identity 1, and let F [x_{1}, …, x_{n}] = F [x⃗] denote the ring of polynomials in variables x_{1}, …, x_{n} with coefficients in F. Let F^{n} denote the set of ordered n-tuples of elements of F.

F^{n} is called an \underline{n-dimensional affine space}. It’s also a vector space, but we won’t need that extra information. Note that F [x⃗] is a commutative ring (it contains the constant polynomials 0 and 1).

Proposition 5.2. F [x_{1}], the polynomial ring in one variable, is a p.i.d.

Proposition 5.3. For n > 1, F [x_{1}, …, x_{n}] is an integral domain, but not a p.i.d.

Exercise 5.4. ℤ[x_{1}] is an integral domain, but not a p.i.d.

Proposition 5.5 (Hilbert Basis Theorem). Every ideal J in F [x⃗] is finitely generated.

Definition 5.6. For any ideal J in F [x⃗], define V(J) = {x⃗ ∈ F^{n} : f(x⃗) = 0 for all f ∈ J}.

Definition 5.7. A subset V ⊂ F^{n} is called a \underline{variety} if there are \underline{finitely many} polynomials f_{1}(x⃗), …, f_{N}(x⃗) ∈ F [x⃗] so that

V = {x⃗ ∈ F^{n} : f_{1}(x⃗) = ⋯ = f_{N}(x⃗) = 0}.

V is also called an “algebraic variety,” or an “affine variety,” or the “solution set” of the system of polynomial equations f_{1} = ⋯ = f_{N} = 0.

Theorem 5.8. For any ideal J in F [x⃗], V(J) is a variety.

Proof. By the Basis Theorem, J = ⟨f_{1}, …, f_{N}⟩. Let V denote the variety defined by these N polynomials. If x⃗ ∈ V(J), then f_{1}(x⃗) = … = f_{N}(x⃗) = 0 since f_{1}, …, f_{N} are all elements of J, and so x ∈ V by definition of V, and V(J) ⊆ V. Conversely, if x⃗ ∈ V, then f_{1}(x⃗) = … = f_{N}(x⃗) = 0, and by Proposition 4.10, any element f ∈ J is of the form

f = ∑_{i=1}^{N} f_{i} · r_{i},

so f(x⃗) = ∑ 0 · r_{i}(x⃗) = 0, and x⃗ ∈ V(J). This shows V ⊆ V(J) and V = V(J).

Theorem 5.9. If J_{1} and J_{2} are ideals in F [x⃗] and J_{1} ⊆ J_{2}, then V(J_{2}) ⊆ V(J_{1}).

Proof. Suppose x⃗ ∈ V(J_{2}), which means that f(x⃗) = 0 for all f ∈ J_{2}. Then, since J_{1} ⊆ J_{2}, f(x⃗) = 0 for all f ∈ J_{1}, and x⃗ ∈ V(J_{1}).

So, the “subset” relation is reversed by the “operation” of forming a variety. The idea is that J_{1} contains fewer polynomials, and its solution set will be larger than the solution set of a system with more polynomial equations that must be satisfied.

Corollary 5.10. If J_{1} = J_{2}, then V(J_{1}) = V(J_{2}).

Proof. This is obvious (from using the previous Theorem on J_{1} ⊆ J_{2} and J_{2} ⊆ J_{1}), but geometrically it means that if {f_{1}, …, f_{N}} and {g_{1}, …, g_{M}} generate the same ideal, then they have the same solution set.

The converse is false: two different ideals can have the same variety.

Example 5.11. Find the solution set of the equations

3x + 5y + 2 = 0
15x + 9y + 6 = 0.

These are two linear polynomials, {f_{1}, f_{2}} in ℚ[x, y]. One approach is to consider this as a linear algebra problem and rewrite it as a matrix equation:

( \begin{matrix}3 & 5 & 2 \\ 15 & 9 & 6 \end{matrix} ) · ( \begin{matrix}x \\ y \\ 1 \end{matrix} ) = ( \begin{matrix}0 \\ 0 \end{matrix} ) .

Finding the solutions (x, y) ∈ ℚ^{2} is equivalent to finding the “kernel” of the matrix.

A different approach to the problem is to multiply the first equation by 5 and then subtract it from the second equation, to get the x terms to cancel, leaving only −16y − 4 = 0. The solution is y = −\frac{1}{4}, and then plugging this into the first equation determines x = −\frac{1}{4}, and (−\frac{1}{4}, −\frac{1}{4}) is the only solution.

In terms of the matrix problem, the “subtracting equations” is really the “row-reduction”

procedure, which corresponds to multiplying the 2 × 3 coefficient matrix by ( \begin{matrix}1 & 0 \\ −5 & 1 \end{matrix} ) on its

left. This simplifies the matrix, without changing the kernel.

In terms of ideals, {f_{1}, f_{2}} generate an ideal J = ⟨f_{1}, f_{2}⟩ ⊆ ℚ[x, y], and V(J) is the solution set. In this case, V(J) is the one-point set \{(−\frac{1}{4}, −\frac{1}{4})\} ⊆ ℚ^{2}. The linear polynomial g = f_{2} −5f_{1} is an element of J, and in fact J = ⟨f_{1}, g⟩. (Since f_{1} ∈ J, and g ∈ J, it follows that ⟨f_{1}, g⟩ ⊆ J. Also, f_{1} and f_{2} = g + 5f_{1} are in ⟨f_{1}, g⟩, so J = ⟨f_{1}, f_{2}⟩ ⊆ ⟨f_{1}, g⟩.) The two approaches to solving the two linear equations, the matrix method, and the subtracting equations method, both started with f_{1}, f_{2}, and found a “simpler” basis {f_{1}, g} for the ideal J. Since ⟨f_{1}, g⟩ is the same ideal as ⟨f_{1}, f_{2}⟩, these ideals have the same variety, but it’s easier to find the solution set when one of the polynomials (g) involves only one variable.

Polynomials f ∈ F [x⃗] define functions F^{n} → F, and, as in Theorem 4.13, F [x⃗] is a subring of the set of all functions M = {f : F^{n} → F }.

Notation 5.12. For the rest of this Section, the symbol I(B) will only be used for subsets B ⊆ F^{n}, and I(B) will always mean the set of polynomials in F [x⃗] which have value 0 at all points in B. This is a special case of the notation introduced in Theorem 4.13, which showed I(B) is an ideal in the polynomial ring F [x⃗].

For example, an ideal J in F [x⃗] defines a subset V(J) ⊆ F^{n}, and Theorem 4.13 says that this subset of F^{n} defines an ideal I(V(J)) in F [x⃗].

Theorem 5.13. J ⊆ I(V(J)).

Proof. If f ∈ J, and x⃗ ∈ V(J), then f(x⃗) = 0 by definition of V(J). So, f(x⃗) = 0 for all x⃗ ∈ V(J), and this is the definition of f ∈ I(V(J)).

Sometimes, I(V(J)) = J, but this is not true in general. Theorem 4.13 also says I(V(J)) has to be a radical ideal, and not all ideals in F [x⃗] are radical. The following Proposition, then, is quite plausible, but it is only stated for F = ℂ, and is not true for F = ℝ or ℚ.

Proposition 5.14 (Hilbert’s Nullstellensatz). If J is an ideal in ℂ[x⃗], then

I(V(J)) = \sqrt{J}.

Theorem 5.15. For a set V ⊆ F^{n}, and a variety W ⊆ F^{n}, V ⊆ W ⟺ I(W) ⊆ I(V).

Proof. The implication V ⊆ W ⟹ I(W) ⊆ I(V) is true for any subsets V, W, and was proved in Theorem 4.13. The other implication starts with assuming I(W) ⊆ I(V). By definition of variety, there are polynomials g_{1}, …, g_{M}, so that W = {x⃗ : g_{1}(x⃗) = ⋯ = g_{M}(x⃗) = 0}. Each g_{i} satisfies g_{i}(x⃗) = 0 for all x⃗ ∈ W, so g_{i} ∈ I(W), and by hypothesis, g_{i} ∈ I(V). The definition of g_{i} ∈ I(V) is that for any v⃗ ∈ V, g_{i}(v⃗) = 0, so g_{1}(v⃗) = ⋯ = g_{M}(v⃗) = 0, and this is the definition of v⃗ ∈ W, which proves V ⊆ W.

Corollary 5.16. For two varieties V, W ⊆ F^{n}, V = W ⟺ I(W) = I(V).

Definition 5.17. An ideal I of the commutative ring R is a \underline{maximal} ideal means: (1) I ≠ R, and (2) if J is an ideal of R and I ⊆ J ⊆ R, then J = I or J = R.

Theorem 5.18. For a point (a_{1}, …, a_{n}) ∈ F^{n}, the ideal

I = ⟨x_{1} − a_{1}, …, x_{n} − a_{n}⟩ ⊆ F [x⃗]

is maximal.

Sketch of Proof. The geometric interpretation is that since the point is a “smallest” non-empty variety, the ideal I({(a_{1}, …, a_{n})}) should be a “largest” ideal strictly contained in F [x⃗]. There is an elementary but technical proof using just algebra.

Proposition 5.19. If I is a maximal ideal in ℂ[x⃗], then there is some point (a_{1}, …, a_{n}) ∈ ℂ^{n} so that

I = ⟨x_{1} − a_{1}, …, x_{n} − a_{n}⟩.

This is a difficult theorem which requires the Nullstellensatz or a similar line of reasoning, so I won’t prove it. It is false if ℂ is replaced by ℝ or ℚ.

Definition 5.20. An ideal I of the commutative ring R is a \underline{prime} ideal means that if a, b ∈ R and a · b ∈ I, then a ∈ I or b ∈ I.

Example 5.21. If p ∈ ℤ is a “prime number,” then pℤ ⊆ ℤ is a prime ideal.

Proposition 5.22. If R is a commutative ring, then the following are equivalent: R is an integral domain ⟺ {0_{R}} is a prime ideal.

(Compare this with Lemma 4.12.)

Proposition 5.23. ⟨f⟩ ⊆ F [x_{1}] is a prime ideal ⟺ f(x_{1}) is an irreducible polynomial in F [x_{1}].

Proposition 5.24. If R is a commutative ring, and I is a maximal ideal, then I is a prime ideal.

Exercise 5.25. If R is a commutative ring, and I is a prime ideal, then I is a radical ideal.

Definition 5.26. A variety V ⊆ F^{n} is \underline{irreducible} means: if V = V_{1} ∪ V_{2} for varieties V_{1}, V_{2}, then V_{1} = V or V_{2} = V.

Theorem 5.27. V is an irreducible variety ⟺ I(V) is a prime ideal in F [x⃗].

Proof. First, suppose V is not irreducible, so there are varieties V_{1} ≠ V, V_{2} ≠ V with V = V_{1}∪V_{2}. By Theorem 3.25, V_{1} ⊆ V ⟹ I(V) ⊆ I(V_{1}). By Corollary 5.16, if I(V) = I(V_{1}) then V_{1} = V, so I(V) ≠ I(V_{1}) and there is some f ∈ I(V_{1}) \ I(V). Similarly, there is some g ∈ I(V_{2}) \ I(V). The product f · g satisfies, for any v⃗ ∈ V, (f · g)(v⃗) = f(v⃗) · g(v⃗) = 0, because either v⃗ ∈ V_{1}, so f(v⃗) = 0, or v⃗ ∈ V_{2}, so g(v⃗) = 0. It follows that f · g ∈ I(V), even though neither f nor g is in I(V), so I(V) is not a prime ideal.

Second, let V = V(⟨h_{1}, …, h_{N}⟩), and suppose I(V) is not a prime ideal, so there are some f, g ∈ F [x⃗] with f ·g ∈ I(V) but f ∉ I(V) and g ∉ I(V). Define varieties V_{1} = V(⟨f, h_{1}, …, h_{N}⟩) and V_{2} = V(⟨g, h_{1}, …, h_{N}⟩). By Theorem 5.9,

{h_{1}, …, h_{N}} ⊆ {f, h_{1}, …, h_{N}} ⟹ ⟨{h_{1}, …, h_{N}}⟩ ⊆ ⟨{f, h_{1}, …, h_{N}}⟩ ⟹ V_{1} ⊆ V.

Because f ∉ I(V), there is some x⃗ ∈ V with f(x⃗) ≠ 0, so x⃗ ∉ V_{1} = V(⟨f, h_{1}, …, h_{N}⟩); this shows V_{1} ≠ V. Similarly, V_{2} ⊊ V, so V_{1} ∪ V_{2} ⊆ V. For any v⃗ ∈ V, h_{1}(v⃗) = … = h_{N}(v⃗) = 0 and because f · g ∈ I(V), (f · g)(v⃗) = 0, so either f(v⃗) = 0 = h_{1}(v⃗) = … = h_{N}(v⃗) ⟹ v⃗ ∈ V_{1}, or g(v⃗) = 0 = h_{1}(v⃗) = … = h_{N}(v⃗) ⟹ v⃗ ∈ V_{2}. In either case, v⃗ ∈ V_{1} ∪ V_{2}, so V ⊆ V_{1} ∪ V_{2} and V = V_{1} ∪ V_{2}: V is not irreducible.

Theorem 5.28. If J is a prime ideal in ℂ[x⃗] then V(J) is an irreducible variety.

Proof. By Proposition 5.14, I(V(J)) = \sqrt{J}, and by Exercise 5.25, because J is a prime ideal, J is also a radical ideal, so I(V(J)) = \sqrt{J} = J. It follows that I(V(J)) is a prime ideal, so V(J) is irreducible by Theorem 5.27.

Example 5.29. Any point in F^{n} is obviously an irreducible variety. It defines a maximal ideal by Theorem 5.18, and this gives a geometric interpretation of Proposition 5.24.

Proposition 5.30. If R is a p.i.d. (recall Definition 4.6), and I is a prime ideal of R, then either I = {0_{R}}, or I is a maximal ideal.

Since F [x_{1}] is a p.i.d., a geometric interpretation of this Proposition is that the only irreducible varieties in F^{1} are sets with one point, or the empty set, or F^{1}. Algebraically, it means that a polynomial with more than one root in F is either reducible in F [x_{1}] (by the Division Theorem), or the constant polynomial 0.

Theorem 5.31. If I and J are ideals in F [x⃗], then V(IJ) = (V(I)) ∪ (V(J)).

Proof. First, to show ⊆, suppose x⃗ ∈ V(IJ), so that k(x⃗) = 0 for all k ∈ IJ, and in particular f(x⃗) · g(x⃗) = 0 for all f ∈ I and g ∈ J. One of the factors must be zero (f(x⃗), g(x⃗) ∈ F, and every field is an integral domain). If f(x⃗) = 0 for all f ∈ I, then x⃗ ∈ V(I). Otherwise, there’s some f so that f(x⃗) ≠ 0, in which case f(x⃗) · g(x⃗) = 0 ⟹ g(x⃗) = 0 for all g ∈ J, so x⃗ ∈ V(J). Either way, x⃗ ∈ V(I) ∪ V(J).

Also, to show the other inclusion, suppose x⃗ ∈ V(I), so that f(x⃗) = 0 for all f ∈ I. Any element in IJ is of the form h = ∑ f ·g, so h(x⃗) = ∑ f(x⃗)·g(x⃗) = ∑ 0·g(x⃗) = 0 ⟹ x⃗ ∈ V(IJ). Similarly, assuming x⃗ ∈ V(J) will imply x⃗ ∈ V(IJ).

Exercise 5.32. If I and J are ideals in F [x⃗], then V(I ∩ J) = (V(I)) ∪ (V(J)).

Hint. Note this is the same union as in the previous Theorem! Use Exercise 3.19, to get IJ ⊆ I ∩ J ⊆ I, and then apply Theorem 5.9. Also, use Theorem 5.9 with I ∩ J ⊆ J.

Exercise 5.33. Let A, B be ideals in F [x⃗]. Show that V(A + B) = (V(A)) ∩ (V(B)).

Hint. Use Theorem 5.9 to show ⊆.

Exercise 5.34. Prove that every prime ideal I in R contains all the nilpotent elements of R.

Hint. This means if a^{n} = 0_{R}, then a ∈ I. One could prove this statement by induction on n, as in Lemma 4.12.

Example 5.35. Let f(x, y) = x^{4}y^{2} + x^{2}y^{4} − 3x^{2}y^{2} + 1 ∈ ℝ[x, y]. Then f(1, 1) = f(1, −1) = f(−1, 1) = f(−1, −1) = 0, and these four points are the only real solutions of f(x, y) = 0, so the real variety is a four-point set in the plane ℝ^{2}:

V = V(⟨f⟩) = {(1, 1), (−1, 1), (1, −1), (−1, 1)}.

There are no other solutions by the arithmetic-geometric mean inequality:

\frac{x^{4}y^{2} + x^{2}y^{4} + 1}{3} ≥ (x^{4}y^{2} · x^{2}y^{4} · 1)^{1/3} = x^{2}y^{2},

with equality only if x^{4}y^{2} = 1 and x^{2}y^{4} = 1.

This V is not irreducible (it is the union V_{1} ∪ V_{2} where V_{1} is a one point variety, and V_{2} is the union of the other three points, which is also a variety).

Comment: this real polynomial f was shown by Motzkin to not be a sum of squares of real polynomials f_{1}^{2} + f_{2}^{2} + … + f_{n}^{2}, even though its values are always ≥ 0.

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