Generalized affine transformation monoids on Galois rings

Abstract

Let A be a ring with identity. The generalized affine transformation monoid Gaff(A) is defined as the set of all transformations on A of the form x↦xu+a (for all x∈A), where u,a∈A. We study the algebraic structure of the monoid Gaff(A) on a finite Galois ring A. The following results are obtained: an explicit description of Green's relations on Gaff(A); and an explicit description of the Schützenberger group of every -class, which is shown to be isomorphic to the affine transformation group for a smaller Galois ring.