α-Skew Linearly McCoy Rings And α-sps McCoy Rings

We have four parts in this paper.The first part: We introduce the grand results in the McCoy ring,α-skew Armendariz ring andα-skew linearly McCoy ring and our main work in this paper.The second part: We investigate some extension properties ofα-skew linearly McCoy ring and the relation betweenα-skew linearly McCoy ring and other rings. The following statement are the main results:Proposition 2.3 If R isα-compatible ring and linearly McCoy ring, R isα- skew linearly McCoy ring.Proposition 2.4 If R isα- skew linearly McCoy ring, the Dorroh extension of R isα- skew linearly McCoy ring.Proposition 2.10 Letαis the endomorphism of R . R isα-rigid ring, then ( )Wn R isα% - skew linearly McCoy ring .Proposition 2.12 Letαis the endomorphism of R ,andαt = IR for some positive integer t ,then R is skew linearly McCoy ring ? R[ x ] isα- skew linearly McCoy ring.The third part: We generalize the concept ofα-Skew McCoy Ring, and pose the concept ofα-sps McCoy ring, and investigate some extension properties ofα-sps McCoy ring. The following statement are the main results:Proposition 3.3 Letαis the endomorphism of R ,u is the central regular element of R .Ifα(u )= u, R isα-sps McCoy ring? uR isα-sps McCoy ring. Proposition 3.5 Let R be a domain, then R isα-sps McCoy ring for any end- omorphismαof R .Proposition 3.7 R isα-compatible ring,αis the endomorphism of R , I < R andα( I )? I. If R I isα- sps Armendariz ring, I is reduced ring, R isα- sps Armendariz ring.The four part: We investigate some properties of matrix ring ofα- skew McCoy ring. The following statement are the main results: Proposition 4.1 Let R be a ring andα,αare the endomorphism of R and S n,α(1 ) = 1,then R isα-skew McCoy ring(?) S n isα-skew McCoy ring.Corollary 4.2α(1 ) = 1,then R isα-skew McCoy ring ? T ( R ,R ) isα-skew McCoy ring.