Consider the ring R = 3Z and the ideal I = 12Z of R. List explicitly all the cosets of I in R and their elements.

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

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

19. Find a specific example of two elements and in a ring such that and .

a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].

15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .

Find the characteristic of each of the following ring: a. b. c. M2() d. M2() e. M2(2) f. M2(3)

If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.

37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.

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.