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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Finitely embedded commutative rings
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by Carl Faith PDF
Proc. Amer. Math. Soc. 112 (1991), 657-659 Request permission

Addendum: Proc. Amer. Math. Soc. 118 (1993), 331.


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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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 112 (1991), 657-659
  • MSC: Primary 13E10; Secondary 16L60, 16P20, 16P60
  • DOI:
  • MathSciNet review: 1057942