Series and 1-affine completeness

Abstract

Given a faithful compatible R-module G of a nearring R, G is a 1-affine complete R-module if R is the same as the nearring $C_0(G)$ of functions on G that are zero and congruence preserving. As a special case, an expanded group is 1-affine complete if this holds when R is the nearring of 0-preserving polynomial functions of the expanded group. The authors have been participants in a series of papers which, when R satisfies the descending chain condition on right ideals, culminate in the determination of the multiplicative group of units of R possessing a faithful compatible R-module G via a series $G = G_0> G_1> \cdots > G_s = 0$ of R-ideals of G where each factor $G_i/G_{i+1}$ is a direct sum of isomorphic minimal R-modules. In this paper we apply work done in this determination of the unit group of R to the study of 1-affine completeness of the R-module G. In particular, we use this work to obtain necessary and sufficient conditions for each factor $G/G_i$ of the preceding series to be a 1-affine complete $R/\mathrm {Ann}_R(G/G_i)$-module. To the authors’ knowledge, every known 1-affine complete group has such a series.