Exercise 17, § 1

Let A be a ring.

(a) Show that if the A-module $A_s^n$ is isomorphic to no $A_s^m$ for $m > n$, then, for all $p < n$, $A_s^p$ is isomorphic to no $A_s^q$ for $q > p$.

(b) Show that if the A-module $A_s^n$ contains no free system of $n + 1$ elements, the A-module $A_s^p$ contains no free system of $p + 1$ elements for $p < n$.