Exercise 23, § 2

(a) Let A be a left Noetherian ring with no divisor of 0. Show that A possesses a field of left fractions (use Exercise 14 (a) to show that condition $(\beta_1)$ of Exercise 22 (b) is fulfilled for $S = A - \{0\}$).

(b) Let $A$ be a left and right Noetherian ring with no divisor of 0. Show that every finitely generated left $A$-module $M$, each of whose elements $\neq 0$ is free, is a submodule of a free $A$-module. (Embed $M$ canonically in $K \otimes_A M$, where $K$ is the field of fractions (left or right) of $A$ considered as a right $A$-module; if $(x_i)$ is a basis of the left vector $K$-space $K \otimes_A M$, show that there exists $a \neq 0$ in $A$ such that $M$ is contained in the sub-$A$-module of $K \otimes_A M$ generated by the $a^{-1} x_i$.)