Rings Over Which Cyclics are Direct Sums of Projective and CS or Noetherian

TL;DR

This paper investigates the structure of rings where cyclic modules decompose into projective and CS or noetherian modules, characterizing when such rings are right noetherian or semisimple artinian.

Contribution

It establishes new conditions under which right WV-rings are right noetherian or semisimple artinian based on module decompositions.

Findings

Right WV-rings are right noetherian iff cyclic modules decompose as specified.

Finitely generated modules over V-rings decompose into semisimple and noetherian parts.

Von Neumann regular rings are semisimple artinian.

Abstract

R is called a right WV -ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV -ring, then R is right uniform or a right V -ring. It is shown that for a right WV-ring R, R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS or noetherian module. For a finitely generated module M with projective socle over a V -ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X \oplus T, where X is semisimple and T is noetherian with zero socle. In the case that M = R, we get R = S \oplus T, where S is a semisimple artinian ring, and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.