Finitely generated mixed modules of Warfield type
Let \(R\) be a local one-dimensional domain which is not a valuation domain. The paper deals with a class of finitely generated mixed modules over \(R\) whose construction is similar to that used by \textit{R. B. Warfield jun.} [Proc. Am. Math. Soc. 25, 167--172 (1970; Zbl 0204.05902)] and called therefore of Warfield type. Theorem 2.1 shows that these \(R\)-modules have local endomorphism rings, hence they are indecomposable. The torsion part \(t(M)\) of a mixed module of Warfield type \(M\) is also examined: torsion elements of \(M\) are characterized in Theorem 3.1, then the author focuses on the natural property \(t(M)\subset \mathfrak{M}M\), where \(\mathfrak{M}\) is the maximal ideal of \(R\). Interestingly, if \(M\) is \(n\)-generated and defined by the elements \(a\) and \(b\) of \(R\), if either \(a/b\) or \(b/a\) is an integral element over \(R\) of degree \(\leq n-1\), then \(t(M)\) is not contained in \(\mathfrak{M}M\) (Theorem 3.3). In the last section of the paper it is shown that \(M/t(M)\) is isomorphic to an ideal of \(R\) and the minimum number of its generators is determined.
scientific article; zbMATH DE number 7367423
scientific article