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Home , Additive identity, Additive inverse, Commutative ring, Domain (ring theory), Homomorphism, Identity element

Chapter 3, Rings

Definitions and examples.

We now have several examples of algebraic systems with and : Z, Zn, R,Mn(R),2Z={2n|n∈Z}. We will write down a system of which includes them all.

Definition, p. 42. A is a nonempty R with two binary operations (usually written as addition and multiplication) such that for all a, b, c ∈ R, (1) R is closed under addition: a + b ∈ R. (2) Addition is associative: (a + b)+c=a+(b+c). (3) Addition is commutative: a + b = b + a. (4) R contains an additive , called zero and usually denoted by 0 or 0R: a +0=0+a=a. (5) Every element of R has an : for each a, there exists an x ∈ R such that a + x =0=x+a. We write x = −a. (6) R is closed under multiplication: ab ∈ R. (7) Multiplication is associative: (ab)c = a(bc). (8) Multiplication distributes over addition: a(b + c)=ab + ac and (a + b)c = ac + bc.

Other possible properties are captured by special types of rings. We will encounter many in this book; many more are studied as well by mathematicians.

Theorem 3.3. Additive inverses are unique.

Proof. Assume that x and y are both inverses of a.Thenx=x+0 = x+(a+y)= (x+a)+y=0+y=y. 

We don’t have to add axioms about . We just define a − b to be a +(−b).

Definition. A is a ring R that satisfies the additional that ab = ba for all a, b ∈ R.

Examples are Z, R, Zn,2Z, but not Mn(R)ifn≥2.

Definition. A ring with identity is a ring R that contains a multiplicative 1R:1Ra=a=a1Rfor all a ∈ R.   10 Examples: 1 in the first three rings above, in M2(R). The set of even 01 1 2

is a ring without identity. The set of odd integers is not a ring. We can also work with matrices whose elements come from any ring we know about, such as Mn(Zr).

Example. LetR = M2(Z2). This is a finite (16 elements) with 10 00 identity 1R = and 0R = . We give an example to show it 01 00 is noncommutative:           10 01 01 01 10 00 = but = 00 00 00 00 00 00

Example: rings of continuous functions. Let X be any topological ; if you don’t know what that is, let it be R or any in R. We consider the set R = C(X, R), the set of all continuous functions from X to R. R becomes a ring with identity when we define addition and multiplication as in elementary : (f + g)(x)=f(x)+g(x)and (fg)(x)=f(x)g(x). The identity element is the constant 1. R is commutative because R is, but it does have zero divisors for almost all choices of X.

There are many, many examples of this sort of ring. The functions don’t have to be continuous. They can be restricted in many other ways, or not restricted at all. For example, you can look at functions or differentiable functions (for some choices of X).

Definition, p. 46. An integral is a commutative ring R with identity 1R =06 R with no zero divisors; that is, ab =0R implies that a =0R or b =0R.

Examples: Z, R, Zp for p prime. Nonexamples: C(R, R), Zn for n composite, the {0R}, the even integers 2Z.

Definition, p. 47. A field is an in which every nonzero element a has a , denoted a−1.

Examples: R, Q, C, Zp for p prime (Theorem 2.8).

If an element of a ring has a multiplicative inverse, it is unique. The proof is the same as that given above for Theorem 3.3 if we replace addition by multiplication. (Note that we did not use the commutativity of addition.) This is also the proof from Math 311 that invertible matrices have unique inverses.

Definition, p. 60. Any element a in a ring R with identity which has an inverse u (i.e., au =1r =ua)iscalledaunit. 3

Making new rings.

Theorem 3.1 ( rings). Let R, S be rings and form the R × S. Define operations by

(r, s)+(r0,s0)=(r+r0,s+s0) (r, s)(r0,s0)=(rr0,ss0).

Then R × S is a ring. If R and S are both commutative, so is R × S.IfRand S both have an identity, then (1R, 1S) is the identity in R × S.

Example. Let R be the ring Z × Z = { (n, m) | n, m ∈ Z }.Notethat(1,0)(0, 1) = (0, 0) = 0R,soRis not an integral domain. For the same reason, no is an integral domain.

Definition, p. 49. A subset of a ring which is itself a ring (using the same operations) is called a . A subset of a field which is itself a field is called a subfield.

Q is a subfield of R, and both are subfields of C. Z is a subring of Q. Z3 is not a subring of Z. Its elements are not integers, but rather are congruence classes of integers. 2Z = { 2n | n ∈ Z } is a subring of Z, but the only subring of Z with identity is Z itself. The zero ring is a subring of every ring.

As with subspaces of vector spaces, it is not hard to check that a subset is a subring as most axioms are inherited from the ring.

Theorem 3.2. Let S be a subset of a ring R. S is a subring of R iff the following conditions all hold: (1) S is closed under addition and multiplication. (2) 0R ∈ S. (3) −s ∈ S for every element s ∈ S.

Proof. Axioms 1, 4, 5 and 6 are in our hypotheses. Axioms 2, 3, 7 and 8 are inherited from R. 

As examples, do exercises from book, page 51–55: 7, 8, 9, 19, 34, discuss 35. 4

Basic properties of rings.

Theorem 3.4–3.5. For all a, b, c in a ring R, (1) a + b = a + c implies b = c. (2) a · 0=0·a=0. (3) a(−b)=−(ab)=(−a)b. (4) −(−a)=a. (5) −(a + b)=(−a)+(−b). (6) (−a)(−b)=ab. (7) (−1)a = −a if R has an identity element.

Proof. These make use of the definition of subtraction and negatives. For (1), add −a. (2) a · 0=a(0 + 0) = a · 0+a·0anduse(1). (3) Show that a(−b)and(−a)bare additive inverses for ab (we know there is a unique such element). (4) and (5) Definition of additive inverse. (6) Use (3) and (4). (7) Use (3) or add (−1)a to a. 

A quicker version of Theorem 3.2 can be obtained using subtraction.

Theorem 3.6. Let R be a ring and ∅6=S⊆R. Sis a subring of R if S is closed under subtraction and multiplication.

Proof. We need to show S is closed under addition, has 0 and has additive inverses. But S =6 ∅ implies there is some s ∈ S, hence 0 = s − s ∈ S. For any a ∈ S, −a =0−a∈S. And finally, for a, b ∈ S, a + b = a − (−b) ∈ S. 

Notation: For a in a ring R and n ∈ Z, write na for a sum of n copies of a and an 0 for a product of n copies of a. This includes the cases 0 · a =0R and a =1R if R has an identity element.

Be careful: (a+b)2 = a2+ab+ba+b2 cannot be simplified unless the ring is commutative.

Theorem 3.7. Let R be a ring and a, b ∈ R. The equation a + x = b has the unique solution x = b − a in R.

Proof. b − a is a solution: check it! If z is another solution, then a + z = b = a +(b−a), so z = b − a by cancellation (Theorem 3.4). 

For multiplication, we need multiplicative inverses to get the cancellation in the proof. 5

Theorem 3.8. Let R be a ring with identity and a, b ∈ R.Ifais a , then the equations ax = b and ya = b have unique solutions in R.

Proof. x = a−1b and y = ba−1 are solutions: check! Uniqueness works as in Theorem 3.7, using the inverse for cancellation: if z is another solution to ax = b,thenaz = b = a(a−1b). Multiply on the left by a−1 to get z = a−1az = a−1a(a−1b)=a−1b. A similar argument works for y. 

The solutions x = a−1b and y = ba−1 may not be the same. Exercise 4, p. 62 gives an example with 2 × 2 matrices.

Sometimes multiplicative cancellation works without inverses (recall the integers).

Theorem 3.10. Let R be an integral domain. If a =06 and ab = ac,thenb=c.

Proof. ab = ac implies a(b − c) = 0. Since a =0and6 Ris an integral domain, we must have b − c =0,orb=c. 

If R is not an integral domain, this fails because of zero divisors:

p. 62. Let a =6 0 in a ring R. a is a if there exists an element b =0in6 R with either ab =0orba =0.

Example: 2 · 3=0=2·0inZ6.

Theorem 3.11. Every finite integral domain is a field.

Combinatorial proof. Let a =6 0 in the integral domain R. The set aR = { ar | r ∈ R } is a of the elements of R: ax = ay implies x = y by Theorem 3.10 and there are only finitely many elements. Therefore some ar must be 1 and a has an inverse. 

Examples. Discuss exercises 8, 11, 17, 25 on pages 62–65. 6

Isomorphisms and .

Recall from linear that a linear transformation is a function between vector spaces that preserves the operations on the . For rings we only want to consider the functions that preserve their two operations.

Definition, p. 71. Let R and S be rings. A function f : R → S (which means the domain is R and f takes values in S)iscalledahomomorphism if f(a + b)=f(a)+f(b)and f(ab)=f(a)f(b) for all a, b ∈ R.

Examples. Z → Q defined by n 7→ n is the natural embedding of the integers into the rational . Z → Zn defined by n 7→ [n]. This is a by the definition of addition and multiplication in Zn. (Theorem 2.6) ιR : R → R, the identity for any ring R. f : C → C defined by f(a + bi)=a−bi (complex conjugation). Check the definition.

The last two examples are special in that they are one-to-one (injective)andonto (surjective). In this case, we say the homomorphism is an .Iff:R→Sis a ring isomorphism, we say R and S are isomorphic. An interesting example in the book is given on p. 69: the complex numbers are isomorphic to the ring of real 2 × 2 matrices ab of the form , which corresponds to the complex a + bi. −ba

Besides the identity, there is one other rather trivial example: the zero mapping z : R → S defined by z(r)=0S for all r ∈ R is a .

Example. Recall Exer. 7, p. 51: R∗ = { (r, r) | r ∈ R }. Define f : R → R∗ by f (r)=(r, r), Check that f is an isomorphism.

Theorem 3.12. Let f : R → S be a ring homomorphism.

(1) f(0R)=0S. (2) f(−a)=−f(a)for all a ∈ R. (3) f(a − b)=f(a)−f(b)for all a, b ∈ R. If R is a ring with identity and f is surjective, then

(4) S is a ring with identity and f(1R)=1S. (5) Whenever u ∈ R is a unit, then f(u) is a unit in S and f(u)−1 = f(u−1).

Proof. (1) f(0) = f(0 + 0) = f(0) + f(0) implies f(0R)=0S. (2) f(−a)+f(a)=f(−a+a)=f(0) = 0S by (1), so f(−a)=−f(a). (3) f(a − b)=f(a+(−b)) = f(a)+f(−b)=f(a)−f(b)by(2). 7

(4) Let s ∈ S. We must show sf(1) = f(1)s = s.Sincefis surjective, there is some r ∈ R with f(r)=s.Thensf(1) = f(r)f(1) = f(r · 1) = f(r)=sand similarly for f(1)s. (5) f(u−1)f(u)=f(u−1u)=f(1) = 1 by (4). 

Note that we really need surjectivity in (4) and (5). If S = R × R and we define f : R → S by f (r)=(r, 0), then f(1) = (1, 0) is not the identity in S. Furthermore, 1 ∈ R is a unit, but f(1) is not a unit.

When the homomorphism f : R → S is not surjective, it is handy to have a name for the subset of S that it maps onto. We write im(f)={f(r)|r∈R}and call this set the of f. (The book does not have a symbol for the image or for the which we shall define later.) Now we can say that f : R → S is surjective if S =im(f).

Corollary 3.13. Given a ring homomorphism f : R → S, the image of f is a subring of S.

Proof. im(f) =6 ∅ since f(0) = 0 ∈ im(f). under subtraction is Theorem 3.12(3). Closure under multiplication follows from the definition of homomorphism: f(a)f(b)= f(ab) ∈ im(f). 

Students should read pages 74–75 for hints on how to decide (or prove) whether two rings are isomorphic or not.

Examples. 1. Z6 is isomorphic to Z2 × Z3. They have the same number of elements. We must find a which preserves the operations. Define f : Z6 → Z2 × Z3 by f ([n]) = ([n]2, [n]3). Check that f is a homomorphism. Since both sets have 6 elements, f will be a bijection if it is either injective OR surjective. Surjectivity is an example of the Chinese Remainder Theorem (page 408). We show injectivity. Assume f([n]) = f([m]). Then f([m − n]) = f([m]) − f([n]) = 0, so it suffices to prove only [0] maps to ([0], [0]). (Recall the situation for vector spaces!) If [a] 7→ ([0]2, [0]3), then a is divisible by both 2 and 3, hence by 6 since they are relatively prime. Therefore [a]=0inZ6.

2. Z4 is not isomorphic to Z2 ×Z2. What fails in trying to use the previous proof is that 2 and 2 are not relatively prime. What is really different about them? Any isomorphism must take 1 ∈ Z4 to (1, 1) ∈ Z2 × Z2.ButinZ4, we must add 1 to itself four times to get zero, while (1, 1) + (1, 1) = (0, 0). Thus any homomorphism f : Z4 → Z2 × Z2 will have f(2) = f(0) = 0 and will not be injective.

These two examples can be generalized. See exercises 39, 40 on page 79.

3. Pages 76–79, exercises 18, 25, 33.