Proposition 4

Let $A$ be a ring which is not necessarily commutative, $m$ a right ideal of $A$ contained in the Jacobson radical of $A$ and $M$ a left $A$-module. Suppose that one of the following conditions holds : (i) $M$ is finitely generated; (ii) $m$ is nilpotent. Then the relation $(A_d/m) \otimes_A M = 0$ implies $M = 0$.

The assertion with respect to hypothesis (i) is precisely Corollary 3 to Proposition 6 of Algebra, Chapter VIII, $§ 6$, no. 3. On the other hand, the relation $(A_d/m) \otimes_A M = 0$ is equivalent to $M = mM$ and hence implies $M = m^nM$ for every integer $n > 0$; whence the assertion with respect to hypothesis (ii).