Weakly left localizable rings

ABSTRACT

A new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S⁻¹R of the ring R. Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals.

KEYWORDS:

• A left localization maximal ring
• a weakly left localizable ring
• denominator set
• the classical left quotient ring of a ring
• the largest left quotient ring of a ring
• the largest regular left Ore set of a ring

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 16U20
• 16P50
• 16S85

Acknowledgement

The author would like to thank the referee for comments and interesting questions. The work is partly supported by the Royal Society and EPSRC.