Exercise 19, § 1

Let A be a non-zero ring with no divisors of zero. (a) Show that for $n > 1$, the A-module $A_s^n$ is not monogenous. (b) Given an integer $n \geq 1$, for the A-module $A_s^n$ to contain no free system of $n + 1$ elements, it is necessary that A admit a field of left fractions (I, § 9, Exercise 15); conversely, if this condition is satisfied, $A_s^n$ contains a free system of $n + 1$ elements for no integer $n > 0$. (Use Exercises 17 and 18 and I, § 9, Exercise 16.) (c) Show that if every left ideal of A is monogenous, A admits a field of left fractions (use (a) and I, § 9, Exercise 16).