On (quasi-)morphic property of skew polynomial rings

Abstract

. The main objective of this paper is to study (quasi-)morphic property of skew polynomial rings. Let R be a ring, σ be a ring homomorphism on R and n ≥ 1. We show that R inherits the quasi-morphic property from R [ x ; σ ] / ( x n +1 ). It is also proved that the morphic property over R [ x ; σ ] / ( x n +1 ) implies that R is a regular ring. Moreover, we characterize a unit-regular ring R via the morphic property of R [ x ; σ ] / ( x n +1 ). We also investigate the relationship between strongly regular rings and centrally morphic rings. For instance, we show that for a domain R , R [ x ; σ ] / ( x n +1 ) is (left) centrally morphic if and only if R is a division ring and σ ( r ) = u − 1 ru for some u ∈ R . Examples which delimit and illustrate our results are provided.