An Essential Extension with Nonisomorphic Ring Structures II

Abstract

We construct a ring R with R = Q(R), the maximal right ring of quotients of R, and a right R-module essential extension S _R of R _R such that S has several distinct isomorphism classes of compatible ring structures. It is shown that under one class of these compatible ring structures, the ring S is not a QF-ring (in fact S is not even a right FI-extending ring), while under all other remaining classes of the ring structures, the ring S is QF. We demonstrate our results by an application to a finite ring.

Key Words:

• (FI-) extending
• Compatible ring structure
• Injective hull
• Kasch ring
• Osofsky compatible ring
• QF-ring

2000 Mathematics Subject Classification:

• Primary 16D50, 16L60
• Secondary 16S50

ACKNOWLEDGMENTS

The authors thank the referee and Professor B. L. Osofsky for their helpful suggestions and comments for improving the results of this article. Also the authors are grateful for the support they received from the Mathematics Research Institute, Columbus and for the kind hospitality and support of Busan National University, the Ohio State University at Lima, and the University of Louisiana at Lafayette.

Notes

Communicated by T. Albu.