Algebra II - August 2023

Instructions

You have 120 minutes to complete this exam.

Problem 1.

(10 points) Let $R$ be an integral domain containing a field $K$ as a subring. Suppose that $R$ is a finite dimensional vector space over $K$ under the ring multiplication. Prove that $R$ is a field.

Problem 2.

(10 points) Let $A$ be a ring with identity. Let

be a short exact sequence of left $A$-modules. Prove that if $L$ and $N$ are both finitely generated, then $M$ is also finitely generated.

Problem 3.

(10 points) Let $A=\mathbb{Q}[x,x^{-1}]$ be the ring of Laurent polynomials. Find an $A$-module which is torsion free but not free.

Problem 4.

(10 points) Let $A$ be a nontrivial commutative ring with identity. Suppose that every ideal in $A$ is free as an $A$-module. Prove that $A$ is a principal ideal domain.

Problem 5.

(10 points) Let $A$ be a ring and let $\alpha \in A$ be an element such that ${\alpha }^2=\alpha$. Prove that the left ideal

is a projective $A$-module.

Problem 6.

(10 points) Compute the Tor group