Warfield domains
Warfield domains are commutative integral domains \(R\) for which, given any \(R\)-submodule \(A\) of the field of quotient \(Q\) of \(R\), all \(A\)-torsionless \(\text{End}_R (A)\)-modules of finite rank are \(A\)-reflexive. This notion has evolved from investigations of R. A. Warfield on abelian groups. The article under review aims at characterizing more generally Warfield domains, since recently results have been obtained for Dedekind and later on for special noetherian integral domains. The extended results read as follows: (1) For a noetherian domain every ideal is two generated if and only if it is a Warfield domain. (2) If \(R\) is integrally closed, then \(R\) is a Warfield domain if and only if \(R\) is an almost maximal strongly discrete Prüfer domain. The proofs use the important characterization of Warfield domains as totally reflexive domains (theorem 6.6). For its proof plenty of results concerning \(A\)-reflexive, \(A\)-divisorial and local divisorial domains are necessary.
scientific article; zbMATH DE number 954785