additional structures on tensor products

LEMMA 9.4   Let $A$ be a (not necessarily commutative) ring. Let $M$ be a right $A$-module. Let $N$ be a left $A$-module. If $M$ carries a structure of an $A$-algebra, then the tensor product $M\times_A N$ carries a structure of $M$-module in the following manner.

$$x. (y\otimes n )= (x y) \otimes n \qquad (x,y\in M, n \in N)$$