Finitely embedded commutative rings

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 2 -sided ace ⊥ \bot rings. A corollary is a result of Shizhong [Sh] that shows commutative subdirectly irreducible acc ⊥ \bot rings are QF.