# Homomorphisms of twisted modules

Let $A$ and $B$ be $R$-bimodules, and $\alpha$ an $R$-automorphism. Write ${}_1A_\alpha$ for the right-twisted $R$-bimodule with action $(r\cdot x\cdot s)\mapsto rx\alpha(s)$.

Is it true that a homomorphism $A\xrightarrow{\varphi}B$ is also a homomorphism ${}_1A_\alpha\xrightarrow{\varphi}{}_1B_\alpha$?

I'm asking this question becasue I do not have much time to find it for myself, and is really just a fact (if it's true) that I'd benefit from. I certainly think it is true, but it's all too easy for me to miss something, like a necessary constraint (or even a rather trivial counterexample that it isn't true in general). Is there a reference for this?

Since you're twisting just on the "right" structure, we might as well just think about right $R$-modules $A,B$, along with an $R$-automorphism $\alpha:R\to R$. Let $A_\alpha, B_\alpha$ be the restrictions of $A,B$ along $\alpha$, and $\varphi:A\to B$ an $R$-module homomorphism. Since $\varphi(a\cdot r)=\varphi(a)\cdot r$, for all $a\in A, r\in R$, we have $$\varphi(a\cdot \alpha(r))=\varphi(a)\cdot \alpha(r)\,,$$ for all $a\in A, r\in R$. This makes the (restricted) map $\varphi: A_\alpha\to B_\alpha$ an $R$-module homomorphism as well.