Baer module hulls of certain modules over a Dedekind domain

Abstract

The notion of a Baer ring, introduced by Kaplansky, has been extended to that of a Baer module using the endomorphism ring of a module in recent years. There do exist some results in the literature on Baer ring hulls of given rings. In contrast, the study of a Baer module hull of a given module remains wide open. In this paper we initiate this study. For a given module [Formula: see text], the Baer module hull, [Formula: see text], is the smallest Baer overmodule contained in a fixed injective hull [Formula: see text] of [Formula: see text]. For a certain class of modules [Formula: see text] over a commutative Noetherian domain, we characterize all essential overmodules of [Formula: see text] which are Baer. As a consequence, it is shown that Baer module hulls exist for such modules over a Dedekind domain. A precise description of such hulls is obtained. It is proved that a finitely generated module [Formula: see text] over a Dedekind domain has a Baer module hull if and only if the torsion submodule [Formula: see text] of [Formula: see text] is semisimple. Further, in this case, the Baer module hull of [Formula: see text] is explicitly described. As applications, various properties and examples of Baer hulls are exhibited. It is shown that if [Formula: see text] are two modules with Baer hulls, [Formula: see text] may not have a Baer hull. On the other hand, the Baer module hull of the [Formula: see text]-module [Formula: see text] ([Formula: see text] a prime integer) is precisely given by [Formula: see text]. It is shown that infinitely generated modules over a Dedekind domain may not have Baer module hulls.