## 1999 March

Instructions: Do as many problems as you can. You are not expected to do all of the problems. You may use earlier parts of a problem to solve later parts, even if you cannot solve the earlier part. Most importantly, give careful solutions. Ph.D. candidates should not do the starred problems.

1.* Suppose $D$ is a commutative ring (not necessarily with 1) having no zero divisors. Give a definite construction of a field in which $D$ can be embedded along with the embedding. You don’t need to prove anything beyond that the operations are well defined and that the set of your construction is closed under the operations.

1. a. Every Euclidean domain is a principal ideal domain.
b. Every principal ideal domain satisfies the ascending chain condition for ideals.
c. Give an example of a unique factorization domain that is not a principal ideal domain.

2. Every finitely generated module over a principal ideal domain is the direct sum of its torsion submodule and a free module.

3. Let $R$ be a ring.
a. Suppose $A$ is a right $R$-module and $B$ is a left $R$-module. Define the Abelian group $A\otimes_R B$.
b. Show that $A\otimes_R R$ and $A$ are isomorphic Abelian groups.

4. Let $I$ be an infinite set. Suppose for each $i\in I$ that $F_i$ is a field. $\prod(F_i : i\in I)$ denotes the direct product of the family of fields, and $\sum (F_i : i\in I)$ denotes the direct sum, and the latter set consists of those sequences belonging to the direct product that have $0$ in all but finitely many places. Show that the ring $\prod(F_i : i\in I)/\sum (F_i : i\in I)$ has a homomorphic image which is a field. (Any assertion you make about the existence of an ideal should be proved.)