Trivial extension of π-regular rings

الملخص EN

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.