Exercise 18, § 1

Let A be a ring for which there exists an integer $p$ with the following property: for every family $(a_i)_{1 \leq i \leq p}$ of $p$ elements of A, there exists a family $(c_i)_{1 \leq i \leq p}$ of elements of A, one at least of which is not a right divisor of 0, such that $\sum_{i=1}^p c_i a_i = 0$.

(a) Show, by induction on $n$, that for every family $(x_i)$ of $p^n$ elements of $A_s^n$, there exists a family $(c_j)$ of $p^n$ elements of $A$, one at least of which is not a right divisor of 0, such that $\sum_j c_j x_j = 0$.

(b) Deduce from (a) and Exercise 16 that for all $n > 0$, the A-module $A_s^n$ contains no free system of $n + 1$ elements.