On Some Classes of Module Hulls

Abstract

The study of various types of hulls of a module has been of interest for a long time. Our focus in this paper is to present results on some classes of these hulls of modules, their examples, counter examples, constructions and their applications. Since the notion of hulls and its study were motivated by that of an injective hull, we begin with a detailed discussion on classes of module hulls which satisfy certain properties generalizing the notion of injectivity. Closely linked to these generalizations of injectivity, are the notions of a Baer ring and a Baer module. The study of Baer ring hulls or Baer module hulls has remained elusive in view of the underlying difficulties involved. Our main focus is to exhibit the latest results on existence, constructions, examples and applications of Baer module hulls obtained by Park and Rizvi. In particular, we show the existence and explicit description of the Baer module hull of a module N over a Dedekind domain R such that N / t(N) is finitely generated and \(\text {Ann}_R(t(N))\ne 0\), where t(N) is the torsion submodule of N. When N / t(N) is not finitely generated, it is shown that N may not have a Baer module hull. Among applications, our results yield that a finitely generated module N over a Dedekind domain is Baer if and only if N is semisimple or torsion-free. We explicitly describe the Baer module hull of the direct sum of \(\mathbb {Z}\) with \(\mathbb {Z}_p\) (p a prime integer) and extend this to a more general construction of Baer module hulls over any commutative PID. We show that the Baer hull of a direct sum of two modules is not necessarily isomorphic to the direct sum of the Baer hulls of the modules, even if each relevant Baer module hull exists. A number of examples and applications of various classes of hulls are included.