Modules With Descending Chain Conditions on Endoimages
Theophilus Gera, Manoj Kumar Patel, Ashok Ji Gupta
TL;DR
The paper develops the theory of endoartinian modules and rings, introducing endoartinianity as a finiteness condition controlled by endomorphism images and connecting it to isoartinian, endonoetherian, and generalized Fitting concepts. It establishes a Hopkins–Levitzki-type equivalence in the commutative and principal-injective settings, and provides structural classifications for semiprime endoartinian rings, including a deep link to Köthe rings via principal ideal rings with central idempotents. It shows that endoartinianity coincides with a Köthe structure in PIRs and yields a decomposition into finite products of artinian uniserial rings, thereby unifying several classical finiteness conditions under the endo-theoretic lens. The work also supplies prime and semiprime corollaries, localization stability results, and explicit examples that delineate the necessity of the stated hypotheses, contributing to a cohesive framework for understanding finiteness conditions in noncommutative ring and module theory.
Abstract
We investigate endoartinian modules, which satisfy the descending chain condition on endoimages, and establish new characterizations that unify classical and generalized chain conditions. Over commutative rings, endoartinianity coincides with rings satisfying the strongly ACCR* with dim(R) = 0 and strongly DCCR* conditions. For principally injective rings, the endoartinian and endonoetherian rings are equivalent. Addressing a question of Facchini and Nazemian, we provide a condition under which isoartinian and Noetherian rings coincide, and we classify semiprime endoartinian rings as finite products of matrix rings over a division ring. We further show that endoartinianity is equivalent to the Kothe rings over principal ideal rings with central idempotents, and characterize such rings as finite products of artinian uniserial rings.