Baer module hulls of certain modules over a Dedekind domain (Q2816907)

The authors, in this quite nice and interesting paper, study the existence of Baer module hull (as a generalization of injective hull of a module) for modules over a Dedekind domain. In the introduction, the authors provide a detailed history of the development of the study of various types of ring hulls and module hulls for various types of rings and various types of modules over special rings. In the rest of this review, unless otherwise specified, \(R\) will stand for a Dedekind domain. The authors' main theorem in this paper is that if \(N\) is an \(R\)-module with \(N/t(N)\) finitely generated and \(\mathrm{ann}_R(t(N))\neq 0\) (where \(t(N)\) is the torsion submodule of \(N\)) (in particular, if \(N\) is finitely generated), \(N\) has a Baer hull if and only if \(t(N)\) is semisimple. Here, I add as a remark that if \(M\) is a finitely generated torsion module over a commutative integral domain \(R\) then \(\mathrm{ann}_R(M)\neq 0\), and hence if \(M\) is a finitely generated module over a commutative Noetherian integral domain \(R\) then \(\mathrm{ann}_R(t(M))\neq 0\). For the proof of the main theorem Theorem 2.13 is crucial. The authors then deduce that a finitely generated \(R\)-module \(N\) is Baer if and only if either \(N\) is semisimple or torsion-free. Finally, the authors give some applications. They give an example of an \(R\)-module which has no Baer hull. If \(\mathfrak B(N)\) denotes the Baer hull of an \(R\)-module \(N\) (when it exists) and if \(N_1\), \(N_2\) are \(R\)-modules admitting Baer hulls \(\mathfrak B(N_1)\), \(\mathfrak B(N_2)\) respectively, though \(N_1\cong N_2\Rightarrow\mathfrak B(N_1)\cong\mathfrak B(N_2)\), the authors give an example to show that the reverse implication not necessarily holds. They also prove that if \(N_1\), \(N_2\) are \(R\)-modules such that \(\mathfrak B(N_1)\), \(\mathfrak B(N_2)\) and \(\mathfrak B(N_1\oplus N_2)\) exist, then \(\mathfrak B(N_1)\oplus\mathfrak B(N_2)\) and \(\mathfrak B(N_1\oplus N_2)\) are not necessarily isomorphic. To conclude, I make the following observation. The authors have used the fact that simple \(\mathbb Z\)-modules are cancellative. For this, it should have been better to give a reference as, I feel, this is not a well-known result.