Left Quotient Associative Pairs and Morita Invariant Properties

Abstract

In this paper, we prove that left nonsingularity and left nonsingularity plus finite left local Goldie dimension are two Morita invariant properties for idempotent rings without total left or right zero divisors. Moreover, two Morita equivalent idempotent rings, semiprime and left local Goldie, have Fountain–Gould left quotient rings that are Morita equivalent too. These results can be obtained from others concerning associative pairs. We introduce the notion of (general) left quotient pair of an associative pair and show the existence of a maximal left quotient pair for every semiprime or left nonsingular associative pair. Moreover, we characterize those associative pairs for which their maximal left quotient pair is von Neumann regular and give a Gabriel-like characterization of associative pairs whose maximal left quotient pair is semiprime and artinian.

Key Words:

• Associative pair
• Left quotient pair
• Morita invariant property
• Fountain–Gould left quotient ring

Mathematics Subject Classification:

• 16S90
• 16D90

Acknowledgments

The authors would like to thank Pere Ara for his comments about Sec. 4. Partially supported by the MCYT, BFM2001-1938-C02-01 and the “Plan Andaluz de Investigación y Desarrollo Tecnológico”, FQM 264.

Notes

^#Communicated by R. Wisbauer.