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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Modules over coproducts of rings

Author: George M. Bergman
Journal: Trans. Amer. Math. Soc. 200 (1974), 1-32
MSC: Primary 16A64
MathSciNet review: 0357502
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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 r gl$ \dim R = {\sup _\Lambda }($r gl$ \dim {R_\lambda })$ if this is > 0, and is 0 or 1 in the remaining case.

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Keywords: Coproduct (or ``free product") of rings, category of $ R$-modules, global dimension
Article copyright: © Copyright 1974 American Mathematical Society