On overrings of Gorenstein Dedekind domains (Q2861451)

The authors study Gorenstein Dedekind domains and their overrings. The concept of Gorenstein Dedekind domains was introduced in [\textit{N. Mahdou} and \textit{M. Tamekkante}, Arab. J. Sci. Eng., 36, No. 3, 431--440 (2011; Zbl 1219.13009)]. Let \(R\) be a commutative ring. An \(R\)-module \(M\) is said to be G-projective if there exists an exact sequence of projective modules NEWLINE\[NEWLINE \mathbf{P}=\cdots\rightarrow P_1\rightarrow P_0\rightarrow P^0\rightarrow P^1\rightarrow\cdots NEWLINE\]NEWLINE such that \(M\cong\text{Im}(P_0\rightarrow P^0)\) and such that \(\text{Hom}_R(-,Q)\) leaves the sequence \(\mathbf{P}\) whenever \(Q\) is a projective \(R\)-module. A domain is called a Gorenstein Dedekind domain (G-Dedekind for short) if every submodule of a projective module is G-projective.NEWLINENEWLINEThe main result in the first section:NEWLINENEWLINETheorem 1.10. Let \(R\) be a one-dimensional Noetherian domain with quotient field \(K\) and integral closure \(T\). Denote the ideal \((R:_K T)\) by \(I\). Then \(R\) is a G-Dedekind domain if and only if for any prime ideal \(P\) of \(R\) which contains \(I\), \(P\) is G-projective.NEWLINENEWLINEAn overring of a domain \(R\) is a ring between \(R\) and its quotient field \(K\). It is well known that any overring of a Dedekind domain is still a Dedekind domain. In Section 2, the authors give the example the overring of the G-Dedekind domain such that this overring is not G-Dedekind.NEWLINENEWLINEA domain \(R\) is called a Warfield domain if, given any submodule \(A\) of the field of quotients \(K\), the \(A\)-torsionless \(\text{End}(A)\)-modules of finite rank are \(A\)-reflexive. The authors provide an example to show that G-Dedekind domains are not necessarily Noetherian Warfield domains and give a definition for a special kind of Noetherian local domain (2-DVR). A Noetherian local domain \(R\) is called 2-DVR if the maximal ideal \(M\) of \(R\) is strongly G-projective (i.e. there exist an exact sequence of projective modules NEWLINE\[NEWLINE \mathbf{P}=\cdots\rightarrow P\rightarrow P\rightarrow P\rightarrow P\rightarrow\cdots NEWLINE\]NEWLINE such that all these projective modules and all the morphisms of the exact sequence are the same and \(M\cong \text{Im}(P\rightarrow P)\) and also, \(\text{Hom}_R(-,Q)\) leaves the sequence \(\mathbf{P}\) exact whenever \(Q\) is a projective module). The authors prove:NEWLINENEWLINETheorem 2.27. Let \(R\) be a Noetherian domain. Then \(R\) is a Warfield domain if and only if, for any maximal ideal \(M\) of \(R\), \(R_M\) is a 2-DVR.