## 2008 April

1. Let $R$ be a commutative ring.
a. Define what it means for $R$ to be Noetherian.
b. Prove that if $R$ is Noetherian, the polynomial ring $R[x]$ is Noetherian.

2. Let $R$ be a commutative ring with identity. Recall that an ideal $P\subset R$ is prime if $P\neq R$ and $ab \in P \Rightarrow a\in P$ or $b\in P$. Let $J\subseteq R$ be an ideal.
a. Show that $J$ is prime if and only if $R/J$ is an integral domain.
b. Show that $J$ is maximal if and only if $R/J$ is a field.
c. Prove that if $J\neq R$, then $J$ is contained in a maximal ideal.
d. Prove that an element of $R$ is nilpotent if and only if it belongs to every prime ideal of $R$.

3. Let $R$ be a ring with identity.
a. Suppose that $A$ is a right $R$-module and $B$ is a left $R$-module. Define the abelian group $A\otimes_R B$.
b. Show that $\mathbb Z_m \otimes \mathbb Z_n \cong \mathbb Z_d$, where $d = \gcd(m,n)$.