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 ) are Artinian. The proof uses the author's characterization in [Fl] of acc rings, and the Levitzki [L] and Herstein-Small [H-S] theorem on the nilpotency of nil ideals in -sided ace rings. A corollary is a result of Shizhong [Sh] that shows commutative subdirectly irreducible acc rings are QF.

**[C]**Victor P. Camillo,*Commutative rings whose quotients are Goldie*, Glasgow Math. J.**16**(1975), no. 1, 32–33. MR**0379476****[F]**Carl Faith,*Lectures on injective modules and quotient rings*, Lecture Notes in Mathematics, No. 49, Springer-Verlag, Berlin-New York, 1967. MR**0227206****[Fl]**Carl Faith,*Rings with ascending condition on annihilators*, Nagoya Math. J.**27**(1966), 179–191. MR**0193107****[F2]**Carl Faith,*Algebra. II*, Springer-Verlag, Berlin-New York, 1976. Ring theory; Grundlehren der Mathematischen Wissenschaften, No. 191. MR**0427349****[F3]**-,*Injective modules over Levitzki rings*, Part I of Injective modules and injective quotient rings, Lecture Notes in Pure and Appl. Math., vol. 72, Marcel Dekker, New York, 1982.**[F4]**-,*Annihilators, associated prime ideals, and Kasch-McCoy quotient rings of commutative rings*, Robert Warfield Memorial Issue, Comm. Algebra (to appear).**[H-S]**I. N. Herstein and L. Small,*Nil rings satisfying certain chain conditions*, Canad. J. Math.**16**(1964), 771–776. MR**0166220****[L]**J. Levitzki,*On nil subrings*, Israel J. Math.**1**(1963), 215–216. MR**0163931****[S-S]**William Schelter and Lance W. Small,*Some pathological rings of quotients*, J. London Math. Soc. (2)**14**(1976), no. 2, 200–202. MR**0429973****[Sh]**P. Shizhong,*Commutative subdirectly irreducible rings with the acc on annihilators are*QF, Comm. Algebra (to appear).**[V]**Peter Vámos,*Classical rings*, J. Algebra**34**(1975), 114–129. MR**0382250**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1991-1057942-0

Article copyright:
© Copyright 1991
American Mathematical Society