let $M$ be an $R$-module and let $S$ be an $R$-algebra through the ring homomorphism $\phi$.
I can make $M\otimes S$ into a $R$-module in several different ways. Either by defining
$r. (m\otimes s)=rm\otimes \phi(r)s$ or
$r. (m\otimes s)=rm\otimes s$ or
$r. (m\otimes s)=m\otimes \phi(r)s$
Here the last structure agrees with the structure given on $M\otimes _S S$ (multiplication in second component) by extension of scalars. What can be said about the relationships among these $R$-modules? In particular in my class it was stated that making $M\otimes _S S$ to an $R$-module by extension of scalars agrees with "the original structure", i.e 3 should be isomorphic with 1 or 2.