Indecomposable Modules and Gelfand Rings

Abstract

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R _P is an Artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings. It is shown that each commutative ring R with a Hausdorff and totally disconnected maximal spectrum is local-global. Moreover, if R is arithmetic, then R is an elementary divisor ring.

Key Words:

• Arithmetic ring
• Clean ring
• Gelfand ring
• Indecomposable module
• Local-global ring
• Totally disconnected space

Mathematics Subject Classification:

• Primary 13C05
• Secondary 13A99

Notes

Communicated by A. Facchini.