Ring Theory Exam

Reading time: about 3 minutes (720 words).

2003 November

1. Let $R$ be a commutative ring, and let $N$ be the set of all nilpotent elements of $R$. Prove that
a. $N$ is an ideal of $R$, and
b. $R/N$ has no nonzero nilpotent elements.

2. a. Show that $\mathbb Z[\sqrt{-1}]$ is a unique factorization domain (UFD).
b. Show that $\mathbb Z[\sqrt{-3}]$ is not a UFD.

3. Let $R$ be a commutative ring with unit. Show that if $R$ contains an idempotent element $e$, then there exist ideals $S$, $T$ of $R$ such that $R = S \oplus T$.

4. Prove that if $R$ is a commutative ring and $I, J$ are ideals of $R$, then there is an $R$-module isomorphism $R/I \otimes R/J \cong R/(I+J)$.

5. Let $M$ be a left $R$-module and $x\in M$. Let $\mathrm{ann}(x) = \{r \in R : rx = 0\}$.
a. Show that $\mathrm{ann}(x)$ is a left ideal of $R$.
b. Prove that there is an $R$-module isomorphism $Rx \cong R/\mathrm{ann}(x)$.