On the Splitting Property of Rings with Restricted Class of Injectivity Domains

Abstract

There has been a growing research interest in exploring rings whose modules are either injective or far from being injective to a specified degree. As is commonly observed in each of such scenarios, rings can be broken as a direct sum, where one of the two components is a semisimple Artinian ring and the other is an indecomposable ring having zero or (essential) homogeneous right socle. Although this phenomenon has become increasingly expected and less surprising with every new study, proving it has never become easier, and in most cases, it has been obtained as a result of considerable efforts. This work aims to bring such studies to a common denominator and tries to explain why this phenomenon occurs each time we deal with the same type of problem. Mathematics Subject Classification. Primary 16D50; Secondary 16S90, 16N60