Abstract:

We investigated the rings with  semicommutative endomorphisms, referred to as the α-sc ring, by means of introducing the notion of semicommutative endomorphisms. A endomorphism α of a ring R is called semicommutative if α(a)b=0 implies  aRα(b)=0 for any a,b∈R.  We discussed the relations between αsc rings  and related rings  and present some extensions of αsc rings. It is proved
 that: 1) if α is a endomorphism of a reduced ring R, then R is αsc if and only if R［x］/〈xⁿ〉 is α-sc, where 〈xⁿ〉is the ideal by xⁿ generated in R and n is any positive integer;  2) if α is an automorphism of R, R is a symmetric right Ore ring, then R is αsc if and only if the classical right quotient ring Q(R) of R is α-sc.

Key words: semicommutative endomorphism, α-sc ring, α-shifting ring, extensions of ring