3.1 Basics

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Throughout this section, all rings shall be assumed to be commutative and

to have (multiplicative) identity. I shall denote the identity element by 1.

Let R be a commutative ring (I’ll probably be redundant for awhile),

and let 0 6= a 2 R. If there exists b 2 R, b 6= 0 such that ab = 0, we say

that a is a zero-divisor . (Thus, b is also a zero-divisor.) If R has no zero

divisors, then R is called an integral domain .

Let R be a ring and let I _ R be an ideal. We call I a prime ideal if

whenever a, b 2 R and ab 2 I, then one of a or b is in I. The following is

basic.

Proposition 3.1.1 I is a prime ideal if and only if the quotient ring R/I

is an integral domain.

If I _ R is an ideal not properly contained in any other proper ideal,

then I is called a maximal ideal . The following is easy.

Lemma 3.1.2 A maximal ideal is always prime.

Proposition 3.1.3 I is a maximal ideal if and only if the quotient ring

R/I is a field.

Let R be a ring and let a 2 R. The set (a) = {ra| r 2 R} is an ideal in

R, called the principal ideal generated by a. Sometimes we shall write Ra

72

3.1. BASICS 73

(or aR) in place of simply writing (a) if we want to emphasize the ring R.

More generally, if a1, a2, . . . , ak 2 R, we shall denote by (a1, a2, . . . , ak) the

ideal {

P

riai| r1, r2, . . . , rk 2 R}.

Exercises

1. Assume that the commutative ring R has zero divisors, but only finitely

many. Prove that R itself must be finite. (Hint: Let a 2 R be a

zero divisor and note that each non-zero element of (a) is also a zero

divisor. Now consider the homomorphism of additive abelian groups

R ! (a), r 7! ra. Every non-zero element of the kernel of this map is

also a zero divisor. Now what?)

2. For which values of n is Z/(n) an integral domain?

3. Prove that if R is a finite integral domain, then R is actually a field.

4. Prove that if R is an integral domain, and if x is an indeterminate over

R, then the polynomial ring R[x] is an integral domain.

5. Let R be a commutative ring and let I, J _ R be ideals. If we define

I + J = {r + s| r 2 I, s 2 J},

IJ = {

X

risi| ri 2 I, si 2 J},

then I + J and IJ are both ideals of R. Note that IJ _ I \ J.

6. Again, let R be a commutative ring and let I, J be ideals of R. We

say that I, J are relatively prime (or are comaximal) if I + J = R.

Prove that if I, J are relatively prime ideals of R, then IJ = I \ J.

7. Prove the Chinese Remainder Theorem: Let R be a commutative ring

and let I, J be relatively prime ideals of R. Then the ring homomorphism

R ! R/I × R/J given by r 7! ([r]I , [r]J ) determines an

isomorphism

R/(IJ) _= R/I × R/J.

More generally, if I1, I2, . . . , Ir _ R are pairwise relatively prime,

then the ring homomorphism R ! R/I1 × R/I2 × · · · × R/Ir, r 7!

([r]I1 , [r]I2 , . . . , [r]Ir ) determines an isomorphism

R/(IJ) _= R/I1 × R/I2 × · · · × R/Ir.

74 CHAPTER 3. ELEMENTARY FACTORIZATION THEORY

8. Let P _ R be a prime ideal, and let I, J _ R be ideals with IJ _ P.

If I 6_ P, prove that J _ P.

9. Let R be a commutative ring and let I _ R be an ideal. If I _

P1 [ P2 [ · · · [ Pr, where P1, P2, . . . Pr are prime ideals, show that

I _ Pj for some index j. (Hint: use induction on r.)

10. Residual Quotients. Let R be a commutative ring and let I, J _ R be

ideals. Define the residual quotient of I by J by setting

I : J = {c 2 R| cJ _ I}.

(a) Prove that I : J is an ideal of R.

(b) Prove that I _ I : J.

(c) Prove that (I : J)J _ I; in fact, I : J is the largest ideal K _ R

satisfying KJ _ I.

(d) For ideals I, J,K _ R, (I : J) : K = I : (JK).

11. Primary Ideals. Let Q _ R be an ideal. We say that Q is primary

if ab 2 Q and a 62 Q implies that bn 2 Q for some positive integer n.

Prove the following for the primary ideal Q _ R:

(a) If P = {r 2 R| rm 2 Q for some positive integer m}, then P is

a prime ideal containing Q. In fact P is the smallest prime ideal

containing Q. (In this case we call Q a P-primary ideal.)

(b) If Q is a P-primary ideal, ab 2 Q, and a 62 P, then b 2 Q.

(c) If Q is a P-primary ideal and I, J are ideals of R with IJ _ Q,

I 6_ P, then J _ Q.

(d) If Q is a P-primary ideal and if I is an ideal I 6_ P, then Q : I =

Q.

12. Suppose that P and Q are ideals of R satisfying the following:

(a) P _ Q.

(b) If x 2 P then for some positive integer n, xn 2 Q.

(c) If ab 2 Q and a 62 P, then b 2 Q.

Prove that Q is a P-primary ideal.

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13. Assume that Q1,Q2, . . .Qr are all P-primary ideals. Show that Q1 \

Q2 \ . . . \ Qr is a P-primary ideal.

14. Let R be a ring and let Q _ R be an ideal. Prove that Q is a primary

ideal if and only if the only zero divisors of R/Q are nilpotent elements.

(An element r of a ring is called nilpotent if rn = 0 for some positive

integer n.)

15. Consider the ideal I = (n, x) _ Z[x], where n 2 Z. Prove that I is a

maximal ideal of Z[x] if and only if n is a prime.

16. If R is a commutative ring and x 2 \{M|M is a maximal ideal}, show

that 1 + x 2 U(R).

76 CHAPTER 3. ELEMENTARY FACTORIZATION THEORY

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