Generalized Semicommutative Rings

We call a ring R generalized semicommutative if, for any a , b ∈ R , ab = 0 only positive integers m and n exist such that a m Rb n = 0. It is shown that the class of generalized semicommutative rings lies in the class of central semicommutative rings and contains the class of weakly semicommutative-I rings, where the inclusions are strict. Relationships between generalized semicommutative rings and rings of other known types have been studied. We present a method for producing generalized semicommutative families from a given generalized semicommutative ring. We also provide several criteria for a generalized semicommutative ring to be reduced.