Exercise 20, § 2

Let $A$ be an integral domain, $B$ an $A$-algebra which is a flat $A$-module and $M$ a torsion-free $A$-module. Show that, if $t \in B$ is not a divisor of zero, the relation $t \cdot z = 0$ for $z \in B \otimes_A M$ implies $z = 0$. (Reduce this to the case where $M$ is finitely generated and, by embedding $M$ in a finitely generated free $A$-module, to the case where $M = A$.)

(*) This is a modification of the terminology "regular ring" introduced in Algebra, Chapter VIII, § 6, Exercise 15, as this has a completely different meaning in Commutative Algebra.