Modules over arbitrary domains
Let R be a commutative ring with a multiplicatively closed set S of non- zero divisors. Suppose for simplicity that the elements in S generate a chain of ideals with trivial intersection. If M is a reduced and torsion- free R-module, let \(\hat M\) denote the S-adic completion of M. Let \(Ines M\) be the set of all endomorphisms of M which map \(\hat M\) into M (homomorphisms of M have unique extensions to \^M). This set is a two- sided ideal of the endomorphism algebra \(End M.\) If A is a reduced and torsion-free R-algebra, we shall construct for any cardinal \(\kappa \geq | A|^{| R|}\) a set of R-modules \(M_ i\) with \(End M_ i=A\oplus Ines M_ i,\quad | M_ i| =\kappa\) and \(Hom(M_ i,M_ j)=Ines(M_ i,M_ j)\) for all \(i\neq j\in 2^{\kappa}.\) This result has consequences concerning decomposition theorems, which are known only in special cases, e.g. if S is countable. In particular, we derive the existence of arbitrary large indecomposable modules over any incomplete valuation domain.
scientific article; zbMATH DE number 3849435
scientific article