Trivial Extension of π-Regular Rings

Abstract

In this paper we investigate if it is possible that the trivial extension ring inherit the properties of the ring and present the relationship between the trivial extension of a ring by an -module and the -regularity of by taking new concepts as -coherent rings and --regular rings which introduced as extensions of the concept of -regularrings. Moreover we studied the possibility of being the trivial extension itself -regular ring according to specific conditions. Thus we proved that if is an Artinian ring, then the trivial extension is a -regular ring. As well as if the trivial extension is a Noetherian -regular ring, then is a -regular ring. On the other hand we showed that if is a field and is an -vector space with infinite dimension, then the trivial extension ring of by is --regular ring.