16.2.12. Let I be an ideal in a commutative ring R. Prove that I[x] is an ideal in R[x]. Prove that R[x]/I[x] = (R/I)[x].

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

Prove that if R is a field, then R has no nontrivial ideals.

Show that the ideal is a maximal ideal of .

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

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.

15. Let and be elements of a ring. Prove that the equation has a unique solution.

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 ].

Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4

8. Prove that the characteristic of a field is either 0 or a prime.