Rings whose modules satisfy chain conditions on essential endosubmodules

Abstract

We study rings R such that all (or certain classes) of right R-modules satisfy chain conditions on essential endosubmodules. It is proved that if all right R-modules have DCC on essential endosubmodules, then R is a right pure semisimple ring, and if all right R-modules have both ACC and DCC on essential endosubmodules, then R is of finite representation type. Rings R such that all right R-modules have ACC on essential endosubmodules are discussed, and it is shown, in particular, that if such a ring R is right Artinian, then R is left pure semisimple.

KEYWORDS:

• Ascending chain condition (ACC)
• descending chain condition (DCC)
• endofinite module
• endomorphism ring
• essential endosubmodule
• pure semisimple ring
• ring of finite representation type

2020 MATHEMATICS SUBJECT CLASSIFICATION Primary:

• 16D70
• 16G10
• 16P70
• Secondary: 16P20

Acknowledgment

The author would like to thank the referee for his/her careful reading of the manuscript and several helpful suggestions.