Let $R$ be a ring and $f:R\rightarrow R$ be a ring homomorphism. ($f(1)=1$)
Let $R$ be given the left operation as the ring operation on $R$ and the right operation as $x•r=xf(r)$, so that $R$ is an $(R,R)$-bimodule.
Let $M$ be a left $R$-module and consider the extension of scalars $R\otimes_R M$.
What is an example of this such that $M$ is not isomorphic to $R\otimes_R M$ as groups?
Above hypothesis implies that $\iota:M\rightarrow R\otimes_R M:x\mapsto 1\otimes x$ is an injective group homomorphism, but I'm not sure whether this can be a surjection
I think $R=F[X]$ ($F$ a field), $f: F[X]\to F[X]$, $X\mapsto 0$ and $M=F[X,Y]/(XY-1)\simeq F[X,X^{-1}]$ is an example. In this case, $R\otimes_R M=0$, while $M\neq0$.