Tilting pair was introduced by Miyashita in 2001 as a generalization of tilting module.In this paper,we construct a tilting left EndΛ(C)-right EndΛ(T)-bimodule for a given tilting pairs(C,T) in modΛ,where Λ is an Artin algebra.
The commuting graph of an arbitrary ring R,denoted byΓ(R),is a graph whose vertices are all non-central elements of R,and two distinct vertices a and b are adjacent if and only if ab=ba.In this paper,we investigate the connectivity and the diameter ofΓ(Zn S3).We show thatΓ(Zn S3)is connected if and only if n is not a prime number.IfΓ(ZnS3)is connected then diam(Γ(ZnS3))=3,while ifΓ(Zn S3)is disconnected then every connected component ofΓ(Zn S3)must be a complete graph with same size,and we completely determine the vertice set of every connected component.
A ring R is called linearly McCoy if whenever linear polynomials f(x),g(x) ∈R[x]\{0} satisfy f(x)g(x)=0,then there exist nonzero elements r,s∈R such that f(x)r=sg(x)=0.For a ring endomorphism α,we introduced the notion of α-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring R[x;α] in place of the ring R[x].A number of properties of this generalization are established and extension properties of α-skew linearly McCoy rings are given.
