The fundamental constituents of iteration digraphs of finite commutative rings

Abstract

For a finite commutative ring R and a positive integer k ⩽ 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = a k. Let R = R 1 ⊕ … ⊕ R s , where s > 1 and R i is a finite commutative local ring for i ∈ {1, …, s}. Let N be a subset of {R 1, …, R s } (it is possible that N is the empty set \(\not 0\)). We define the fundamental constituents G * N (R, k) of G(R, k) induced by the vertices which are of the form {(a 1, …, a s ) ∈ R: a i ∈ D(R i ) if R i ∈ N, otherwise a i ∈ U(R i ), i = 1, …, s}, where U(R) denotes the unit group of R and D(R) denotes the zero-divisor set of R. We investigate the structure of G* N (R, k) and state some conditions for the trees attached to cycle vertices in distinct fundamental constituents to be isomorphic.