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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Finitely embedded commutative rings


Author: Carl Faith
Journal: Proc. Amer. Math. Soc. 112 (1991), 657-659
MSC: Primary 13E10; Secondary 16L60, 16P20, 16P60
Addendum: Proc. Amer. Math. Soc. 118 (1993), null.
MathSciNet review: 1057942
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Abstract: A theorem of Ginn and Moss [G-M] states that a right finitely embedded (= with finite essential right socle) two-sided Noetherian ring is Artinian. An example of Schelter and Small [S-S] can be applied to show that the theorem fails for right finitely embedded rings with the ascending chain conditions on right and left annihilators. We show here, however, that finitely embedded commutative rings with the acc on annihilator (= acc $ \bot $) are Artinian. The proof uses the author's characterization in [Fl] of acc $ \bot $ rings, and the Levitzki [L] and Herstein-Small [H-S] theorem on the nilpotency of nil ideals in $ 2$-sided ace $ \bot $ rings. A corollary is a result of Shizhong [Sh] that shows commutative subdirectly irreducible acc $ \bot $ rings are QF.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1057942-0
PII: S 0002-9939(1991)1057942-0
Article copyright: © Copyright 1991 American Mathematical Society