Categories of modules and the correct “language”

Let $R$ be a ring, two $R$-modules are said to be isomorphic if there exists a linear bijective map between the modules. But the first assumption is the two modules are over the same ring. Is it possible to generalize the term of isomorphic modules into the next case?

Let $S$ ($S\ncong R$) be a ring, $M$ be a $R$ module, and $N$ be a $S$ module; if $M$ is isomorphic as groups to $N$, then of course $M$ heirs a structure of a $S$ module and $N$ heirs a structure of a $R$ module. But although $S\ncong R$, is it possible to think of $M\cong N$ as modules in some categories?

For example, if $M=\Bbb{C}[x,y] \big /\langle x^2+y^2-1 \rangle$ is a $R=\Bbb{C}[x+y]$ module, of course $M\cong \Bbb{C}[x,x^{-1}]$ as groups (moreover as rings but it doesn’t matter), is it true now to think of $M$ as a $\Bbb{C}[x+x^{-1}]$ module?, if not, are there some categories such that $M\cong \Bbb{C}[x,x^{-1}]$ as modules?

Solution 1:

What is the structure of $M=C[x,y]/(x^2+y^2-1)$ as a $C[x+y]$-module? Well, in the first place $M$ being a ring implies is a $C[x,y]/(x^2+y^2-1)$-module. Then the ring homomorphisms $C[x+y]\hookrightarrow C[x,y]\twoheadrightarrow C[x,y]/(x^2+y^2-1)$ allow you to "pullback" the $C[x,y]/(x^2+y^2-1)$-module structure to a $C[x+y]$-module structure.

You can use this idea to define module homomorphisms between modules over different rings. Namely, if $M$ is an $R$-module, and $N$ is an $S$-module, then a homomorphism from $M$ to $N$ will be a ring homomorphism $R\to S$, allowing you to "pullback" the $S$-module structure of $N$ into an $R$-module structure of $M$, together with a map $M\to N$ that is an $R$-module homomorphism from $M$ as an $R$-module to $N$ as the $R$-module resulting from "pulling back" the $S$-module structure of $N$ along the ring homomorphism $R\to S$.

An isomorphism from an $R$-module $M$ to an $S$-module $N$ then amounts to an isomorphism $R\cong S$ of the rings, together with an isomorphism of $R$-modules $M\cong N$, where $N$ has the $R$-module structure pulled back along the isomorphism $R\cong S$.