Utumi modules

ABSTRACT

A right R-module M is called a U-module if, whenever A and B are submodules of M with A≅B and A ∩ B = 0, there exist two summands K and L of M such that A⊆^essK, B⊆^essL and K⊕L⊆^⊕M. The class of U-modules is a simultaneous and strict generalization of three fundamental classes of modules; namely, the quasi-continuous, the square-free, and the automorphism-invariant modules. In this paper we show that the class of U-modules inherits some of the important features of the aforementioned classes of modules. For example, a U-module M is clean if and only if it has the finite exchange property, if and only if it has the full exchange property. As an immediate consequence, every strongly clean U-module has the substitution property and hence is Dedekind-finite. In particular, the endomorphism ring of a strongly clean U-module has stable range 1.

KEYWORDS:

• Clean rings and the exchange property
• injective
• quasi-continuous
• quasi-injective
• square-free and automorphism-invariant modules
• U-modules

1991 MATHEMATICS SUBJECT CLASSIFICATION:

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

Acknowledgment

The authors would like to thank the referee for his careful reading of the manuscript, for his suggestions to improve the presentation of the paper, and for providing us with Theorem 5.1 and its proof. This research was supported by the Mathematics Research Institute of the Ohio State University.