Modules Whose Endomorphism Rings are Division Rings

Abstract

The well-known Schur's Lemma states that the endomorphism ring of a simple module is a division ring. But the converse is not true in general. In this paper we study modules whose endomorphism rings are division rings. We first reduce our consideration to the case of faithful modules with this property. Using the existence of such modules, we obtain results on a new notion which generalizes that of primitive rings. When R is a full or triangular matrix ring over a commutative ring, a structure theorem is proved for an R-module M such that End _R (M) is a division ring. A number of examples are given to illustrate our results and to motivate further study on this topic.

Key Words:

• Division ring
• Endomorphism ring
• Rudimentary ring

2010 Mathematics Subject Classification:

• 16K99
• 16S50

ACKNOWLEDGMENTS

The authors are indebted to the referee for valuable suggestions. Especially, the referee pointed out that a right rudimentary domain need not be left or right Ore. The authors are also grateful to Professor S. Tariq Rizvi for his helpful comments and discussions.

Notes

Communicated by T. Albu.