What We Need to Know about Rings and Modules

Class Notes for Mathematics 700 by Ralph Howard

October 31, 1995

Contents

1 Rings 1 1.1 The Definition of a Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Examples of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 The Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 The Ring of Polynomials over a Field . . . . . . . . . . . . . . . . . . . . 3 1.2.3 The Integers Modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Ideals in Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Euclidean Domains 5 2.1 The Definition of Euclidean Domain . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The Basic Examples of Euclidean Domains . . . . . . . . . . . . . . . . . . . . . 5 2.3 Primes and Factorization in Euclidean Domains . . . . . . . . . . . . . . . . . . 6

3 Modules 8 3.1 The Definition of a Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Examples of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 R and its Submodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.2 The Coordinate Space R^{n}. . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.3 Modules over the Polynomial Ring F[x] Defined by a Linear Map. . . . . 10

3.3 Direct Sums of Submodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1 Rings

1.1 The Definition of a Ring

We have been working with fields, which are the natural generalization of familiar objects like the real, rational and complex numbers where it is possible to add, subtract, multiply and divide. However there are some other very natural objects like the integers and polynomials over a field where we can add, subtract, and multiply, but where it not possible to divide. We will call such rings. Here is the official definition: Definition 1.1 A commutative ring (R, +, ·) is a set R with two binary operations + and · (as usual we will often write x · y = xy) so that 1. The operations + and · are both commutative and associative:

x + y = y + x, x + (y + z) = (x + y) + z, xy = yx, x(yz) = (xy)z.

2. Multiplication distributes over addition:

x(y + z) = xy + xz.

3. There is a unique element 0 ∈ R so that for all x ∈ R

x + 0 = 0 + x = x.

This element will be called the zero of R.

4. There is a unique element 1 ∈ F so that for all x ∈ R

x · 1 = 1 · x = x.

This element is called the identity of R.

5. 0 ≠ 1. (This implies R has at least two elements.)

6. For any x ∈ R there is a unique −x ∈ R so that

x + (−x) = 0.

(This element is called the negative of x. And from now on we write x + (−y) as x − y.)

We will usually just refer to “the commutative ring R” rather than “the commutative ring (R, +, ·)”. Also we will often be lazy and refer to R as just a “ring” rather than a “commutative ring”^{1}. As in the case of fields we can view the positive integer n as an element of ring R by setting

n := \underbrace{1 + 1 + ⋯ + 1}_{n terms}

Then for negative n we can set n := −(−n) where −n is defined by the last equation. That is 5 = 1 + 1 + 1 + 1 + 1 and −5 = −(1 + 1 + 1 + 1 + 1).

While in a general ring it is not possible to divide by arbitrary nonzero elements (that is to say that arbitrary nonzero elements do not have inverses as division is defined in terms of multiplication by the inverse), it may happen that there are some elements that do have inverses and we can divide by these elements. We give a name to these elements. Definition 1.2 Let R be a commutative ring. Then an element a ∈ R is a unit or has an inverse b iff ab = 1. In this case we write b = a^{−1}. Thus when talking about elements of a commutative ring saying that a is a unit just means a has an inverse.

1For those of you how can not wait to know: A non-commutative ring satisfies all of the above except that multiplication is no longer assumed commutative (that is it can hold that xy ≠ yx for some x, y ∈ R) and we have to add that both the left and right distributive laws x(y + z) = xy + xz and (y + z)x = yx + zx hold. The natural example of such a non-commutative ring is the set of square n × n matrices over a field with the usual addition and multiplication.

1.2 Examples of Rings

1.2.1 The Integers

The integers Z are as usual the numbers 0, ±1, ±2, ±3, … with the addition and multiplication we all know and love. This is the main example you should keep in mind when thinking about rings. In Z the only units (that is elements with inverses) are 1 and −1.

1.2.2 The Ring of Polynomials over a Field

Let F be a field and let F[x] be the set of all polynomials

p(x) = a_{0} + a_{1}x + a_{2}x^{2} + ⋯ + a_{n}x^{n}

where a_{0}, …, a_{n} ∈ F and n = 0, 1, 2 . . .. These are added, subtracted, and multiplied in the usual manner. This is the example that will be most important to us, so we review a little about polynomials. First if p(x) is not the zero polynomial and p(x) is as above with a_{n} ≠ 0 then n is the degree of p(x) and this will be denoted by n = deg p(x). The the nonzero constant polynomials a have degree 0 and we do not assign any degree to the zero polynomial. If p(x) and q(x) are nonzero polynomials then we have

deg(p(x)q(x)) = deg(p(x)) + deg(q(x)).

Also if given p(x) and f(x) with p(x) not the zero polynomial we can “divide”^{2} p(x) into f(x). That is there are unique polynomials q(x) (the the quotient) and r(x) (the the reminder) so that

f(x) = q(x)p(x) + r(x) where deg r(x) < deg p(x) or r(x) is the zero polynomial.

This is called the division algorithm. If p(x) = x − a for some a ∈ F then this becomes

f(x) = q(x)(x − a) + r where r ∈ F.

By letting x = a in this equation we get the fundamental Proposition 1.3 (Remainder Theorem) If x − a is divided into f(x) then the remainder is r = f(a). If particular f(a) = 0 if and only if x − a divides f(x). That is f(a) = 0 iff f(x) = (x − a)q(x) for some polynomial q(x) with deg q(x) = deg f(x) − 1. I am assuming that you know how to add, subtract and multiply polynomials, and that given f(x) and p(x) with p(x) not the zero polynomial that you can divide p(x) into f(x) and find the quotient q(x) and remainder r(x). Problem 1 Show that the units in R := F[x] are the nonzero constant polynomials.

2Here we are using the word “divide” in a sense other than “multiplying by the inverse”. Rather we mean “find the quotient and remainder”. I will continue to use the word “divide” in both these senses and trust it is clear from the context which meaning is being used.

1.2.3 The Integers Modulo n

This is not an example that will come up often, but it does illustrate that rings can be quite different than the basic example of the integers and the polynomials over a field. You can skip this example with no ill effects. Basically this is a generalization of the example of finite fields. Let n > 1 be an integer and let Z/n be the integers reduced modulo n. That is we consider two integers x and y to be “equal” (really congruent modulo n) if and only if they have the same remainder when divided by n in which case we write x ≡ y mod n. Therefore x ≡ y mod n if and only if x − y is evenly divisible by x. It is easy to check that

x_{1} ≡ y_{1} mod n and x_{2} ≡ y_{2} mod n implies
x_{1} + y_{2} ≡ x_{1} + y_{2} mod n and x_{1}x_{2} ≡ y_{1}y_{2} mod n.

Then Z/n is the set of congruence classes modulo n. It only takes a little work to see that with the “obvious” choice of addition and multiplication that Z/p satisfies all the conditions of a commutative ring. Show this yourself as an exercise.) Here is the case n = 6 in detail. The possible remainders when a number is divided by 6 are 0, 1, 2, 3, 4, 5. Thus we can use for the elements of Z/6 the set {0, 1, 2, 3, 4, 5}. Addition works like this. 3 + 4 = 1 in Z/6 as the remainder of 4 + 3 when divided by 6 is 1. Likewise 2 · 4 = 2 in Z/6 as the remainder of 2 · 4 when divided by 6 is 2. Here are the addition and multiplication tables for Z/6

+ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 4 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4  \begin{matrix}· & 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 \\ 2 & 0 & 2 & 4 & 0 & 2 & 4 \\ 3 & 0 & 3 & 0 & 3 & 0 & 3 \\ 4 & 0 & 4 & 2 & 0 & 4 & 2 \\ 5 & 0 & 5 & 4 & 3 & 2 & 1 \end{matrix}

This is an example of a ring with zero divisors, that is nonzero elements a and b so that ab = 0. For example in Z/6 we have 3 · 4 = 0. This is different from what we have seen in fields where ab = 0 implies a = 0 or b = 0. We also see from the multiplication table that the units in Z/6 are 1 and 5. In general the units of Z/n are the correspond to the numbers x that are relatively prime to n.

1.3 Ideals in Rings

I can not think of a natural way to motivate this idea, which is unfortunate as it very useful and it is very natural in the sense that once you start doing anything nontrivial with rings ideals force themselves on you. However if for the time being it does not look like a natural idea don’t worry, it gets easier. Definition 1.4 Let R be a commutative ring. Then a nonempty subset A ⊂ R is an ideal if and only if it is closed under addition and multiplication by elements of R. That is

a, b ∈ A implies a + b ∈ A

(this is closure under addition) and

a ∈ A, r ∈ R implies ar ∈ A

(this is closure under multiplication by elements of R). There are two trivial examples of ideals in any R. The set A = {0} is an ideal as is A := R. While it is possible to give large numbers of other examples of ideals in various rings for our work in this class we only really need to know one more example: Problem 2 Let R be a commutative ring and let a ∈ R. Let ⟨a⟩ be the set of all multiples of a by elements of R. That is

⟨a⟩ := {ra : r ∈ R}.

Then show A := ⟨a⟩ is an ideal in R.

Definition 1.5 If R is a commutative ring and a ∈ R, then ⟨a⟩ as defined in the last exercise is the principle ideal defined generated by a.

2 Euclidean Domains

2.1 The Definition of Euclidean Domain

As we said above for us the most important examples of rings are the ring of integers and the ring of polynomials over a field. We now make a definition that captures many of the basic properties these two examples have in common. Definition 2.1 A commutative ring R is a Euclidean domain iff

1. R has no zero divisors^{3}. That is if a ≠ 0 and b ≠ 0 then ab ≠ 0. (Or in the contrapositive form ab = 0 implies a = 0 or b = 0.)

2. Then is a function δ : (R \ {0}) → {0, 1, 2, 3, …} (that is δ maps nonzero elements of R to nonnegative integers) so that

(a) If a, b ∈ R are both nonzero then δ(a) ≤ δ(ab).

(b) The division algorithm holds in the sense that if a, b ∈ R and a ≠ 0 then we can divide a into b to get a quotient q and a reminder r so that

b = aq + r where δ(r) < δ(a) or r = 0

2.2 The Basic Examples of Euclidean Domains

Our two basic examples of Euclidean domains are the integers Z with δ(a) = |a|, the absolute value of a and F[x], the ring of polynomials over a field F with δ(p(x)) = deg p(x). We record this as theorems: Theorem 2.2 The integers Z with δ(a) := |a| is a Euclidean domain.

3In general a commutative ring R with no zero divisors is called an integral domain or just a domain.

Theorem 2.3 The ring of polynomials F[x] over a field F with δ(p(x)) = deg p(x) is a Euclidean domain.

Proofs: These follow form the usual division algorithms in Z and F[x].

2.3 Primes and Factorization in Euclidean Domains

We now start to develop the basics of “number theory” in Euclidean domains. By this is meant that we will show that it is possible to define define things like “primes” and greatest “common divisors” and show that they behave just as in the case of the integers. Many of the basic facts about Euclidean domains are proven by starting with subset S of the Euclidean domain in question and then choosing an element a in S that minimizes δ(a). While it is more or less obvious that it is always possible to do this we record (without proof) the result that makes it all work. Theorem 2.4 (Axiom of Induction) Let N := {0, 1, 2, 3, …} be the natural numbers (which is the same thing as the nonnegative integers). Then any nonempty subset S of N has a smallest element.

We start with some elementary definitions:

Definition 2.5 Let R be a commutative ring. Let a, b ∈ R.

1. Then a is a divisor of b, (or a divides b, or a is a factor of b) iff there is c ∈ R so that b = ca. This is written as a | b.

2. b is a multiple of a iff a divides b. That is iff there is c ∈ R so that b = ac.

3. The element b ≠ 0 is a prime^{4}, also called an irreducible, iff b is not a unit and if a | b then either a is a unit, or a = ub for some unit u ∈ R.

4. The element c of R is a greatest common divisor of a and b iff c | a, c | b and if d ∈ R is any other element of R that divides both a and b then d | c. (Note that greatest common divisors are not unique. For example in the integers Z there both 4 and −4 are greatest common divisors of 12 and 20, while in the polynomial ring R[x] if element the c(x − 1) is a greatest common divisor of x^{2} − 1 and x^{2} − 3x + 2 for any c ≠ 0.)

5. The elements a and b are relatively prime iff 1 is a greatest common divisor of a and b. Or what is the same thing the only elements that divide both a and b are units.

There are commutative rings where some pairs of elements do not have any greatest common divisors. We now show that this is not the case in Euclidean domains.

Theorem 2.6 Let R be a Euclidean domain. Then every ideal in R is principle. That is if I is an ideal in R then there is an a ∈ R so that I = ⟨a⟩. Moreover if {0} ≠ I = ⟨a⟩ = ⟨b⟩ then a = ub for some unit u.

Problem 3 Prove this along the following lines:

4I have to be honest and remark that this is not the usual definition of a prime in a general ring, but is the usual definition of an irreducible. Usually a prime is defined by the property of Theorem 2.9. In our case (Euclidean domains) the two definitions turn out to be the same.

1. By the Axiom of induction, Theorem 2.4, the set S := {δ(r) : r ∈ I, r ≠ 0} has a smallest element. Let a be a nonempty element of I that minimizes δ(r) over nonzero elements of I. Then for any b ∈ I show that there is a q ∈ R with b = aq by showing that if b = aq + r with r = 0 or δ(r) < δ(a) (such q and r exist by the definition of Euclidean domain) than in fact r = 0 so that b = qa.

2. With a as in the last step show I = ⟨a⟩, and thus conclude I is principle.

3. If ⟨a⟩ = ⟨b⟩ then a ∈ ⟨b⟩ so there is a c_{1} so that a = c_{1}b. Likewise b ∈ ⟨a⟩ implies there is a c_{2} ∈ R so that b = c_{2}a. Putting these together implies a = c_{1}c_{2}a. Show this implies c_{1}c_{2} = 1 so that c_{1} and c_{2} are units. Hint: Use that a(1 − c_{1}c_{2}) = 0 and that in a Euclidean domain there are no zero divisors.

Theorem 2.7 Let R be a Euclidean domain and let a and b be nonzero elements of R. Then a and b have at least one greatest common divisor. More over if c and d are both greatest common divisors of a and b then d = cu for some unit u ∈ R. Finally if c is any greatest common divisor of a and b then there are elements x, y ∈ R so that

c = ax + by.

Problem 4 Prove this as follows:

1. Let I := {ax + by : x, y ∈ R}. Then show that I is an ideal of R.

2. Because I is an ideal by the last theorem the ideal I is principle so I = ⟨c⟩ for some c ∈ R. Show that c is a greatest common divisor of a and b and that c = ax + by for some x, y ∈ R. Hint: That c = ax + by for some x, y ∈ R follows from the definition of I. From this show c is a greatest common divisor of a and b.

3. If c and d are both greatest common divisors of a and b then by definition c | d and d | c. Use this to show d = uc for some unit u.

Theorem 2.8 Let R be a Euclidean domain and let a, b ∈ R be relatively prime. Then there exist x, y ∈ R so that

ax + by = 1.

Problem 5 Prove this as a corollary of the last theorem.

Theorem 2.9 Let R be a Euclidean domain and let a, b, p ∈ R with p prime. Assume that p | ab. Then p | a or p | b. That is if a prime divides a product, then it divides one of the factors.

Problem 6 Prove this by showing that if p does not divide a then it must divide b. Do this by showing the following:

1. As p is prime and we are assuming p does not divide a then a and p are relatively prime.

2. There are x and y in R so that ax + py = 1.

3. As p | ab there is a c ∈ R with ab = cp. Now multiply both sides of ax + py = 1 by b to get abx + pby = b and use ab = cp to conclude p divides b.

Corollary 2.10 If p is a prime in the Euclidean domain R and p divides a product a_{1}a_{2} ⋯ a_{n} then p divides at least one of a_{1}, a_{2}, …, a_{n}.

Proof: This follows from the last proposition by a straight forward induction.

Lemma 2.11 Let R be a Euclidean domain. Then a nonzero element a of R is a unit iff δ(a) = δ(1).

Problem 7 Prove this. Hint: First note that if 0 ≠ r ∈ R then δ(1) ≤ δ(1r) = δ(r). Now use the division algorithm to write 1 = aq + r where either δ(r) < δ(a) = δ(1) or r = 0.

Proposition 2.12 Let R be a Euclidean domain and a and b nonzero elements of R. If δ(ab) = δ(a) then b is a unit.

Problem 8 Prove this. Hint: Use the division algorithm to divide ab into a. That is there are q and r ∈ R so that a = (ab)q + r so that either r = 0 or δ(r) < δ(a). Then write r = a(1 − bq) and use that if x and y are nonzero δ(x) ≤ δ(xy) to show (1 − bq) = 0. From this show b is a unit.)

Theorem 2.13 (Fundamental Theorem of Arithmetic) Let a be a non-zero element of a Euclidean domain that is not a unit. Then a is a product a = p_{1}p_{2} ⋯ p_{n} of primes p_{1}, p_{2}, …, p_{n}. Moreover we have the following uniqueness. If a = q_{1}q_{2} ⋯ q_{m} is anther expression of a as a product of primes, then m = n and after a reordering of q_{1}, q_{2}, …, q_{n} there are units u_{1}, u_{2}, …, u_{n} so that q_{i} = u_{i}p_{i} for i = 1, …, n.

Problem 9 Prove this by induction on δ(a) in the following steps.

1. As a is not a unit the last lemma implies δ(a) > δ(1). Let k : min{δ(r) : r ∈ R, δ(r) > δ(1)}. Show that if δ(a) = k than a is a prime. (This is the base of the induction.)

2. Assume that δ(a) = n and that it has been shown that for any b ≠ 0 with δ(b) < n that either b is a unit or b is a product of primes. Then show that a is a product of primes. Hint: If a is prime then we are done. Thus it can be assumed that a is not prime. In this case a = bc where b and c are not units. a is a product a = bc with both b and c not units. By the last proposition this implies δ(b) < δ(a) and δ(c) < δ(a). So by the induction hypothesis both b and c are products of primes. This shows a = bc is a product of primes.

3. Now show uniqueness in the sense of the statement of the theorem. Assume a = p_{1}p_{2} ⋯ p_{n} = q_{1}q_{2} ⋯ q_{m} where all the p_{i}’s and q_{j}’s are prime. Then as p_{1} divides the product q_{1}q_{2} ⋯ q_{m} by Corollary 2.10 this means that p_{1} divides at least one of q_{1}, q_{2}, …, q_{m}. By reordering we can assume that p_{1} divides q_{1}. As both p_{1} and q_{1} are primes this implies q_{1} = u_{1}p_{1} for some unit u_{1}. Continue in this fashion to complete the proof.

3 Modules

3.1 The Definition of a Module

A module is basically a vector space over a ring. While at the level of definitions this seems to be all there is to say, at the deeper level of what theorems hold in vector spaces over fields as opposed to the results that hold for modules over rings they differ a great deal. As a basic example all vector spaces have a basis, while “most” modules over rings do not. One of the main theorems of this course is that finitely generated (i.e. modules spanned by a finite number of elements) over a Euclidean domain have something that is very like a basis.

Definition 3.1 Let R be a commutative ring. Then M is a module over R (or an R-module) iff M has an operation + of addition and a zero element 0 so that

1. The operation + is both commutative and associative. That is

x + y = y + x, x + (y + z) = (x + y) + z

for all x, y, z ∈ M.

2. For all x ∈ M we have 0 + x = x + 0 = x.

3. There is an operation of multiplying “vectors” (elements of M) by “scalars” (elements of the base ring R) so that multiplication distributes over addition. If (r, x) ↦ rx from R × M is the scalar multiplication map (which is just the high brow way of saying that rx denotes the scalar product of r ∈ R and x ∈ M. The distributive laws are then

(r_{1} + r_{2})x = r_{1}x + r_{2}y r(x_{1} + x_{2}) = rx_{1} + ry_{2}.

4. Finally we assume that if 1 is the unit element of R that 1x = x for all x ∈ M.

A submodule of an R module is defined just as a subspace of a vector space is defined.

Definition 3.2 Let R be a module over the commutative ring R. Then A ⊂ M is a submodule iff for all x_{1}, x_{2} ∈ A and r_{1}, r_{2} ∈ R we have r_{1}x_{1} + r_{2}x_{2} ∈ A. That is A is closed under linear combinations where the scalars come out of the ring R.

3.2 Examples of Modules

These are easy to come by:

3.2.1 R and its Submodules.

If R is a commutative ring then we can set M = R in the definition of a module and see that R is a module over itself. A subset A ⊆ R is easily checked to be a submodule iff it is an ideal in R.

3.2.2 The Coordinate Space R^{n}.

Just as in the case of the vector space F^{n} of n tuples over a field F we can start with a commutative ring R and form the space of n tuples of elements of R. That is

R := {(x_{1}, x_{2}, …, x_{n}) : x_{1}, x_{2}, …, x_{n} ∈ R}.

This is an R module with the usual operations:

(x_{1}, x_{2}, …, x_{n}) + (y_{1}, y_{2}, …, y_{n}) = (x_{1} + y_{1}, x_{2} + y_{2}, …, x_{n} + y_{n})
r(x_{1}, x_{2}, …, x_{n}) = (rx_{1}, rx_{2}, …, rx_{n}).

3.2.3 Modules over the Polynomial Ring F[x] Defined by a Linear Map.

Let F be a field and V a vector space over F. Let T : V → V be a linear map from V to V. Using T we can now make V into an module over the ring F[x]. We already know how to add elements of V so we just need to define how to multiple elements of V by elements of F[x]. This is done by the rule

f(x) · v := f(T)v.

That is if f(x) = a_{0}+a_{1}x+⋯+a_{n}x^{n} first plug T into f(x) to get f(T) = a_{0}I +a_{1}T +⋯+a_{n}T^{n} which will also be a linear map f(T) : V → V. Then f(x) · v is the image of v under f(T).

3.3 Direct Sums of Submodules

Definition 3.3 Let M be a module over the commutative ring R and let U_{1}, …, U_{n} be submodules of M. Then U_{1}, …, U_{n} are linearly independent iff

x_{1} + x_{2} + ⋯ + x_{n} = 0 with x_{i} ∈ U_{i} implies x_{1} = x_{2} = ⋯ = x_{n} = 0.

Definition 3.4 Let U_{1}, …, U_{n} be submodules of M where M is a module over the commutative ring R. Then there M is a direct sum of U_{1}, …, U_{n} iff U_{1}+U_{2}+⋯+U_{n} = M and also U_{1}, …, U_{n} are linearly independent. In this case we write either

M = U_{1} ⊕ U_{2} ⊕ ⋯ ⊕ U_{n} or M = ⨁_{i=1}^{n} U_{i}.

Proposition 3.5 If M is an R module and U_{1}, …, U_{n} are submodules so that M = U_{1} ⊕ U_{2} ⊕ ⋯ ⊕ U_{n} then every element x ∈ M can be uniquely expressed as x = x_{1} + x_{2} + ⋯ + x_{n} where x_{i} ∈ U_{i}.

Problem 10 Prove this.