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Abstract

In this paper we investigate if it is possible that the trivial extension ring T(R,R) inherit the properties of the ring R and present the relationship between the trivial extension T(R,M) of a ring R by an R-module M and theπ-regularity of Rby taking new concepts asπ-coherent rings and C-π-regular rings whichintroducedas extensions of the concept of π-regularrings. Moreover we studied the possibility of being the trivial extensionT(R,M) itselfπ-regular ring according to specific conditions. Thus we proved that if R is an Artinian ring, then the trivial extension T(R,R) is a π-regular ring. As well as ifthe trivial extension T(R,R) is a Noetherian π-regularring, then R is a π-regular ring. On the other hand we showed that if F is a field and M is an F-vector space with infinite dimension, then the trivial extension ring T(F,M) of F by M is C-π-regular ring.

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