Change-of-rings tensor quotient ($M{\otimes }_{R}N\twoheadrightarrow M{\otimes }_{A}N$)

For a homomorphism of commutative rings $R\to A$ and two $A$-modules, the tensor product of modules over $A$ is the quotient of the tensor product of modules over $R$ imposing all relations $am\otimes n=m\otimes an$. The quotient map is $A$-linear when $A$ acts through the first factor on the source. This expresses the extra balancing imposed by change of rings.