additional structures on tensor products

LEMMA 09.4   Let $A,B$ be (not necessarily commutative) rings. Let $M$ be a right $A$-module. Let $N$ be a left $A$-module. If $M$ carries a structure of a $B$-algebra (so that $M$ is actually a $B$-$A$-bimodule,) then the tensor product $M\times_A N$ carries a structure of $B$-module in the following manner.

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

Under the asummption of the lemma, we see that:

1. $M\otimes_A N$ is additively generated by $\{m\otimes_A n \vert m\in M,n \in N\}$.
2. $m a \otimes_A n= m \otimes_A a n$ $(\forall m\in M ,\forall n \in N, \forall a \in A )$
3. $M\otimes_A N$ carries the structure of $B$-module.: $b.(m\otimes_A n) = (b.m) \otimes_A n \quad (\forall b \in B, \forall m \in M , \forall n \in N)$