Ibrahim Nonkane: Affine completeness of modules of rank 1 over a Principal Ideal Domain

In this paper we generalize some affine completeness properties of abelian groups to modules over commutative principal ideal domains

1. Introduction
The question of affine completeness of modules over commutative rings has been studied by many authors. The case of abelian groups is well known thanks to W. Nbauer [7], K.Kaarli [3] and A.Saks [8]. As pointed out in [1], most of the results of the affine completeness of abelian groups can be generalized to modules over commutative principal ideal domains since abelian groups and modules over commutative principal ideal domains are similar. There is only one exception according to K.Kaarli and A.Pixley: when proving that an abelian group of rank one with bounded torsion part is not affine complete, the authors relied on the countability of the ring of integers (Theorem 5.2.22 [1]). This argument does not hold if it has to do with a ring which is uncountable. This leads to the following problem raised in [1] (Problem 5.2.29) Problem: Does there exist an affine complete torsion free module of rank 1 over a commutative principal ideal domain?

The aim of this work is to answer this question; moreover we give a generalization of another theorem from abelian group’s affine completeness to modules over a commutative ring.

References
[1] K.Kaarli & A.F.Pixley, Polynomial completeness in Algebraic systems London, New york, Washington (2000).
[2] K.Kaarli, Compatible function extension property, Algebra Universalis 17 (1983) 200-207.
[3] K.Kaarli, Affine complete Abelian groups, Math. Nachr. 107 (1982), 235-239.
[4] G. Kientega & I. Rosenberg, Extension of partial operations and relations, Math. sci. Res.J., vol.8(2004), 12, 362-372.
[5] L.Fuchs, Infinite abelian groups, I,II, New York and London 1970, 1973. [6] S.Lang, Algebra, Springer-Verlag 2002.
[7] W.Nbauer, Über die affinvollständige Modulen, Math. Nachr. 86, (1978), 85-96.
[8] A.Saks, On affine completeness of decomposable modules Tartu lik. Toimetoised 764, (1985) 123-135.