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Proceedings of the American Mathematical Society

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

 

 

Ring extensions and essential monomorphisms


Author: Tilmann Würfel
Journal: Proc. Amer. Math. Soc. 69 (1978), 1-7
MSC: Primary 16A33; Secondary 16A56
DOI: https://doi.org/10.1090/S0002-9939-1978-0482574-6
MathSciNet review: 482574
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Abstract: We study pairs of rings $ R \subset S$ such that $ \operatorname{Hom}_R(S, - ):R - \operatorname{Mod} \to S - \operatorname{Mod}$ preserves essential monomorphisms. We obtain a complete characterization of such a pair in case S is a torsion-free algebra over a Noetherian domain $ R \ne \mathrm{Quot}(R)$; S is then a left ideally finite R-algebra. The rings R such that every ring extension $ R \subset S$ satisfies the above condition are subdirect sums of certain Artinian rings. Furthermore, we study a generalization of trivial ring extensions and show that the center of a semi-Artinian ring is again semi-Artinian.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0482574-6
Keywords: Ring extensions, essential monomorphisms, injective hulls, descent of injectivity, Noetherian modules, ideally finite algebras, semi-Artinian rings
Article copyright: © Copyright 1978 American Mathematical Society