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- W1996195439 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 âstandardâ $R$-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." @default.
- W1996195439 created "2016-06-24" @default.
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- W1996195439 date "1974-01-01" @default.
- W1996195439 modified "2023-10-15" @default.
- W1996195439 title "Modules over coproducts of rings" @default.
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- W1996195439 doi "https://doi.org/10.1090/s0002-9947-1974-0357502-5" @default.
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