Rings all of whose right ideals are U-modules

ABSTRACT

The notion of a U-module was introduced and thoroughly investigated in [11] as a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper a right R-module M is called a U*-module if every submodule of M is a U-module, and a ring R is called a right U*-ring if R_R is a U*-module. We show that M is a U*-module iff whenever A and B are submodules of M with A≅B and A∩B = 0, A⊕B is a semisimple summand of M; equivalently M = X⊕Y, where X is semisimple, Y is square-free, and X & Y are orthogonal. In particular, a ring R is a right U*-ring iff R is a direct product of a square-full semisimple artinian ring and a right square-free ring. Moreover, right U*-rings are shown to be directly-finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable-range 1, and hence is stably-finite. These results are non-trivial extensions of similar ones on rings all of whose right ideals are either quasi-continuous or auto-invariant.

KEYWORDS:

• Directly-finite
• injective
• quasi-continuous
• quasi-injective
• square-free and automorphism-invariant modules
• stably finite and exchange rings
• U-modules

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D40, 16D50, 16D60
• Secondary: 16L30, 16L60, 16P20, 16P40, 16P60

Acknowledgments

The authors would like to thank the referee for the careful reading of the manuscript and the suggestions to greatly improve the presentation of the paper. This research was supported by the Mathematics Research Institute of the Ohio State University.