So suppose we have a $R$-$S$-bimodule $M$, and an $S$-$T$ bimodule $N$, then we can construct the abelian group $M\otimes_S N$. Under what conditions could we make this a module?
I would expect this to be the case if $R=S=T$, but are there weaker conditions which would do the job?
As it has been mentioned in the above comments, $M\otimes_SN$ is an $R$-$T$ bimodule with the following actions; let $r\in R$ , $t\in T$ and $m\otimes n\in M\otimes_SN$ ( a pure tensor) we have $$r.(m\otimes n)=r.m\otimes n$$ and $$(m\otimes n).t=m\otimes(n.t)$$ these actions are also compatible, since $r.((m\otimes n).t)=r.(m\otimes(n.t))=(r.m)\otimes(n.t)=(r.(m\otimes n)).t$