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Independence in completions and endomorphism algebras. (English) Zbl 0691.13004

Let R be a commutative ring with identity and let A be an R-algebra. The so-called realization problem asks for the construction of an R-module M with endomorphism algebra \(End_ RM=A\). Significant progress has been made on this problem since A. L. S. Corner’s fundamental paper in Proc. Lond. Math. Soc., III. Ser. 13, 687-710 (1963; Zbl 0116.024) which is largely due to S. Shelah’s work on combinatorial set theory. The current article addresses the problem as to when an R-algebra A of small cardinality can be realized as endomorphism algebra of a module M of the same (or not much larger) cardinality.
The main result is as follows: Let S be a multiplicatively closed set of non-zero divisors of R such that \(\cap_{S}Rs =0\), let A be an S- cotorsion-free R-algebra and let \(\kappa\) be a cardinal. If \(| S| =\aleph_ 0\leq | A| \leq \kappa <2^{\aleph_ 0}\) then there exists an R-module M with \(| M| =\kappa\) and \(End_ RM=A\). This answers a question due to Corner and fills a cardinality gap in realization theorems due to A. L. S. Corner and R. Göbel [Proc. Lond. Math.Soc., III. Ser. 50, 447-479 (1985; Zbl 0562.20030)]. For the special case that \(R={\mathbb{Z}}[X]\) where X is a set of commuting indeterminates and S consists of all monomials in R, the authors show that the topology on R induced by S is complete if and only if X is uncountable.
Reviewer: J.Hausen

MSC:

13B10 Morphisms of commutative rings
13B35 Completion of commutative rings
16S50 Endomorphism rings; matrix rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
20K20 Torsion-free groups, infinite rank
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
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