Exercise 11.2. Let R be a ring. Show that for every nilpotent element a = R (i.e. an = 0 for some nЄ N), the polynomial 1 – ax Є R[x] is a unit in R[x].

18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .

24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)

If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]

14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.

Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.

Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.

Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .

[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [Type here]

Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?