A study on (m, n)-centralizer finite rings

Abstract

For any m distinct elements r 1 ,r 2 , ··· ,r m in a ring R , the m -element central-izer of { r 1 ,r 2 , ··· ,r m } in R is defined as C R ( { r 1 ,r 2 , ··· ,r m } ) = { s ∈ R | sr 1 = r 1 s,sr 2 = r 2 s,...,sr m = r m s } , where m ∈ N with m ⩾ 2. We define the set of all distinct m -element centralizers in a ring R by m − Cent( R ), where m ∈ N with m ⩾ 2. Further, R is called ( m,n )-centralizer ring if | m − Cent( R ) | = n , where n ∈ N . In this paper, we characterize some of the ( m,n )-centralizer finite rings for n ⩽ 7.