Exercise 25, § 4

Generalise Th. 1 of VII, p. 18 to left modules over a ring $A$ which is not necessarily commutative, in which every left ideal and every right ideal is principal (an example of such a ring is the tensor product $K[X] \otimes_K D$, where $K$ is a commutative field and $D$ is a $K$-algebra which is a field but not necessarily commutative).