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.
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The authors thank the referee for a number of suggestions that improved the presentation and completeness of this paper.
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Portions of this paper were stimulated by discussions between the authors while both were guests of Johannes Kepler Universität Linz in Austria in May, 2019. The authors thank the University for its hospitality and are pleased to acknowledge partial support from the Austrian Science Fund FWF (P29931) for this visit.
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Peterson, G.L., Scott, S.D. Series and 1-affine completeness. Rend. Circ. Mat. Palermo, II. Ser 72, 1251–1275 (2023). https://doi.org/10.1007/s12215-022-00726-x
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DOI: https://doi.org/10.1007/s12215-022-00726-x