# Homomorphisms of representations

Given an abelian group $M$ and a ring $R$. Then $(M,\rho)$ is called a representation of $R$, if $\rho:R\rightarrow\text{End}(M)$ is a ring homomorphism.

There is a bijection $$\{\sigma:R\times M\rightarrow M|(M,\sigma) \text{ is a left R-module}\}\rightarrow\{\rho:R\rightarrow\text{End}(M)|(M,\rho)\text{ is a representation of }R\}$$ if we send $\sigma\mapsto\rho$ with $\rho(r)=\sigma(r,\cdot)$.

My question is, how the notion of a module homomorphism can be transported to representations?

• The same way. See for example here for linear representations. – Dietrich Burde Oct 24 '16 at 19:00
• Thanks! In which way this is "transportet"? – user369147 Oct 24 '16 at 19:39

If $\sigma:R\times M\to M$ and $\tau:R\times N \to N$ correspond to $\rho:R\to\mathrm{End}(M)$ and $\psi:R\to \mathrm{End}(N)$, then a module homomorphism $f:M\to N$ is a group homomorphism such that $f\circ \rho(r)=\psi(r)\circ f$ for all $r\in R$.