Morphic property of the quotient rings of a skew polynomial ring

The main goal of this research article is to study the morphic property of the quotient ring of a skew (left) polynomial ring in [Formula: see text] by [Formula: see text]. We prove that if [Formula: see text] is an endomorphism of a ring [Formula: see text] and [Formula: see text] then [Formula: see text] is von Neumann regular if [Formula: see text] is left morphic in [Formula: see text]. Using this result, it is shown that a ring [Formula: see text] is unit regular if [Formula: see text] is left morphic. Further, we prove that for any endomorphism [Formula: see text] of a ring [Formula: see text], the ring [Formula: see text] is left morphic and [Formula: see text] is abelian if and only if [Formula: see text] is strongly regular and [Formula: see text] for all [Formula: see text]. Also it is shown that for any endomorphism [Formula: see text] of a ring [Formula: see text], the ring [Formula: see text] is left centrally morphic if and only if [Formula: see text] is strongly regular and there exists [Formula: see text] such that [Formula: see text] for all [Formula: see text].