Exercise 1, § 7

Let $A$ be an integral domain and $K$ its field of fractions. (a) Show that the exterior power $\bigwedge^2 K$ of the $A$-module $K$ reduces to 0. (b) For every $A$-module $E \subset K$, show that every exterior power $\bigwedge^p E$ is a torsion $A$-module for $p \geq 2$.

(c) If E is the A-module defined in II, § 7, Exercise 31, show that $\bigwedge^2 E$ is non-zero. Give an example of an integral domain A and an A-module E contained in the field of fractions K of A such that no exterior power $\bigwedge^p E$ reduces to 0.