Digraphs of the kTH power mapping over some finite commutative rings

In this dissertation, we consider a local extension of the Galois ring of the form GR(pᵑ,d)[x]/(ʄ(x)ͣ), where n d and a are positive integers p is a prime and ʄ(x) is a monic polynomial in GR(pᵑ,d)[x]/ of degree r such that the reduction in is irreducible. We establish the exponent of R without completely determination of its unit group structure. We obtain better analysis of the iteration graphs G(k) ® induced from the k-th power mapping including the conditions on symmetric digraphs. In addition, we work on the digraph over a finite chain ring R. The structure of G(k)2 such as indeg (k) 0 and maximum distance for G(k)2 ® is determined by the nilpotency of maximal ideal M of R.