## 1996 April

1. Identify the following rings (integral domain, unique factorization domain, principle ideal domain, Dedekind domain, Euclidean domain, etc), citing theorems to justify your answers and giving examples to show why it is not of a more specialized class.
a. $\mathbb F_2[x,y]$
b. $\mathbb F_2[x,y]/(y^2 + y + 1)$
c. $\mathbb R[x,y]/(x^2+y^2)$
d. $\mathbb C[x,y]/(x^2+y^2)$
e. $\mathbb Z[\sqrt{-5}]$

2. a. Describe the ring $\mathbb Q \otimes_{\mathbb Z} \mathbb Q$.
b. Generalize the result of part a. to $\mathbb Q \otimes_{\mathbb Z} A$, where $A$ is a divisible abelian group. Note: the group $A$ (written additively) is called divisible if for any integer $n\geq 2$ and any element $a\in A$, there exists an element $b\in A$ such that $nb = a$.

3. a. Show that any commutative ring has a maximal ideal.
b. Give an example of a commutative ring with only one maximal ideal.
c. A ring as in b. is called a local ring. Show that a commutative ring is local if and only if the set of nonunits forms an ideal.

4. a. Let $R$ be a commutative ring and $M$ an $R$-module. Show that $\operatorname{Hom}_R(R,M)$ and $M$ are isomorphic as left $R$-modules.
b. It is clear that $\operatorname{Hom}R(R,M) \subseteq \operatorname{Hom}{\mathbb Z}(R,M)$. Give an explicit example of a $\mathbb Z$-module homomorphism which is not an $R$-module homomorphism.

5. Let $M$ be a module over an arbitrary ring $R$.
a. Define what it means for $M$ to be projective.
b. Prove that $M$ is projective if and only if it is a direct summand of a free $R$-module.