additional structures on tensor products

LEMMA 3.4   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.

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