## Ring Theory Exam

Reading time: less than a minute (191 words). ## 2000 November 10

1. Let $I$ be an ideal of a ring $R$.
a. Describe (without proofs) a one-to-one correspondence between the set of ideals of $R$ containing $I$ and the set of ideals of $R/I$.
b. Prove that the correspondence matches prime ideals to prime ideals.
c. Prove that an ideal $I$ of a commutative ring $R$ is maximal if and only if $R/I$ is a field.

2. Prove that every finite integral domain is a field.

3. If $F$ is a field, prove that the polynomial ring $F[x]$ is a principal ideal domain. Is the same true for $\mathbb Z[x]$? Why?

4. If $D$ is a principal ideal domain, but not a field, then prove that $D$ satisfies the ascending chain condition for ideals.

5. a. If $I$ and $J$ are ideals in a commutative ring $R$ such that $I + J = R$, then prove that $IJ = I \cap J$.
b. With $R$ commutative, prove that the set of non-units forms an ideal of $R$ if and only if $R$ contains a unique maximal ideal, and give an example of such an $R$.