Modules over coproducts of rings

Abstract:

Let ${R_0}$ be a skew field, or more generally, a finite product of full matrix rings over skew fields. Let ${({R_\lambda })_{\lambda \in \Lambda }}$ be a family of faithful ${R_0}$rings (associative unitary rings containing ${R_0}$) and let $R$ denote the coproduct ("free product") of the ${R_\lambda }$ as ${R_0}$-rings. An easy way to obtain an $R$-module $M$ is to choose for each $\lambda \in \Lambda \cup \{ 0\}$ an ${R_\lambda }$-module ${M_\lambda }$, and put $M = \oplus {M_\lambda }{ \otimes _{{R_\lambda }}}R$. Such an $M$ will be called a “standard” $R$-module. (Note that these include the free $R$-modules.) We obtain results on the structure of standard $R$-modules and homomorphisms between them, and hence on the homological properties of $R$. In particular: (1) Every submodule of a standard module is isomorphic to a standard module. (2) If $M$ and $N$ are standard modules, we obtain simple criteria, in terms of the original modules ${M_\lambda },{N_\lambda }$, for $N$ to be a homomorphic image of $M$, respectively isomorphic to a direct summand of $M$, respectively isomorphic to $M$. (3) We find that $\text {r gl} \dim R = {\sup _\Lambda }(\text {r gl}\dim {R_\lambda })$ if this is > 0, and is 0 or 1 in the remaining case.