Commuting Pair Preservers of Matrices

Abstract. There are many papers on linear operators that preserve commuting pairs ofmatrices over fields or semirings. From these research works, we have a motivation to theresearch on the linear operators that preserve commuting pairs of matrices over nonneg-ative integers. We characterize the surjective linear operators that preserve commutingpairs of matrices over nonnegative integers.1. Introduction and preliminariesLet Z + be the nonnegative part of the ring of integers Z and let M n (Z + ) denote theset of all n × n matrices over Z + . Similarly let B = {0,1} be the binary Boolean algebraand let M n (B) denote the set of all n × n matrices over B. We denote the n × n identitymatrix by I n and the n×n zero matrix by O n . The n×n matrix all of whose entries arezero except its (i,j)th, which is 1, is denoted E i,j . We call E i,j a cell. We denote the n×nmatrix all of whose entries are 1 by J n . We omit the subscripts on I,O, and J when theyare implied by the context. If A and B are matrices in M