Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that:
For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$
Examples of this bimodules can be seen as follows.
If $R$ is a commutative ring, $M$ is a central $R$-$R$ bimodule (that is $mr=rm,$ for all $m\in M,r\in R$) and $G$ is a group acting on $R$ by automorphisms. Then, for any $g\in G$ we can construct a new $R$-$R$-bimodule $_gM_I,$ where .
$_gM_I=M$ as sets and the operations of $R$ on $M$ are $r\cdot m=g(r)m$ and $m\bullet r=mr.$ Then we see that $r\cdot m=g(r)m=mg(r)=m\bullet g(r), $ that is, $$r\cdot m=m\bullet g(r).$$
I want to know if there is a name for this kind of bimodules.
Thanks.