When is Every Module with Essential Socle a Direct Sums of Quasi-Injectives?

ABSTRACT

In the present article we study the structure of rings, over which essential extensions of semisimple modules are direct sums of quasi-injectives. In the special case of commutative rings, these rings are precisely Artinian PIR and so every module over such rings is a direct sum of cyclics as characterized by Köthe and Cohen-Kaplansky.

Key Words:

• Artinian
• Essential submodule
• PIR
• Q.F.D.
• Quasi-injective
• Socle

1991 Mathematics Subject Classification:

• 16D70
• 16P20

ACKNOWLEDGMENTS

S. K. Jain would like to dedicate this paper in the memory of Kostia who has left a vacuum in Ring Theory. The authors express their gratitude to V. Uspenskiy for bringing Engelking (1977) to their attention.

Notes

^†Kostia Beidar passed away on March 8, 2004.

Communicated by A. Facchini.