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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Exact embedding functors and left coherent rings


Authors: Kent R. Fuller and George Hutchinson
Journal: Proc. Amer. Math. Soc. 104 (1988), 385-391
MSC: Primary 16A89; Secondary 18E20
DOI: https://doi.org/10.1090/S0002-9939-1988-0962803-9
MathSciNet review: 962803
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Abstract: Let $ R$ and $ S$ be rings with unit. Suppose $ P$ is a free $ R$-module on $ \beta $ generators, where $ \beta $ is an infinite cardinal number not smaller than the cardinality of $ R$, and $ T$ is the ring of endomorphisms $ \operatorname{End}{(_R}P)$.

Theorem. If $ R$ is left coherent and there exists an exact embedding functor $ F:R - \operatorname{Mod} \to S - \operatorname{Mod}$, then $ _SF{(R)_R}$ is a bimodule such that $ F{(R)_R}$ is faithfully flat.

Theorem. If $ F:R - \operatorname{Mod} \to S - \operatorname{Mod}$ is an exact embedding functor, then $ _R{P_T}$ is a bimodule such that $ _RP$ is a projective generator (inducing an exact embedding Hom functor from $ R - \operatorname{Mod}$ into $ T - \operatorname{Mod}$,) and $ _SF{(T)_T}$ is a bimodule such that $ F{(T)_T}$ is faithfully flat (inducing an exact embedding tensor product functor $ _SF(T){ \otimes _T}$ -- from $ T - \operatorname{Mod}$ into $ S - \operatorname{Mod}$.)

Theorem. There exists an exact embedding functor $ R - \operatorname{Mod} \to S - \operatorname{Mod}$ iff there exists an $ S$-module $ N$ and a unit-preserving ring monomorphism $ h:\operatorname{End}{(_R}P) \to \operatorname{End}{(_S}N)$ of their endomorphism rings, such that $ h$ preserves and reflects exact pairs of endomorphisms.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0962803-9
Article copyright: © Copyright 1988 American Mathematical Society

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