Irreducibility in the Total Ring of Quotients

Abstract

Let R be a ring whose total ring of quotients Q is von Neumann regular. We investigate the structure of R when it admits an ideal that is irreducible as a submodule of the total ring of quotients. We characterize those rings which contain a maximal ideal that is irreducible in its total ring of quotients Q. An integral domain has a Q-irreducible ideal which is a maximal ideal if and only if R is a valuation domain. We show that when the total ring of quotients of R is von Neumann regular, then having a maximal ideal that is Q-irreducible is equivalently to certain valuation like properties, including the property that the regular ideals are linearly ordered.

Key Words:

• Irreducible
• P-primal
• Total ring of quotients

2000 Mathematics Subject Classification:

• Primary 13A15, 13F05

Notes

Communicated by I. Swanson