Prescribing endomorphism algebras, a unified treatment. (English) Zbl 0562.20030
This highly interesting, well-written and timely paper applies Saharon Shelah’s ingenious Black Box to the simultaneous investigation of endomorphism algebras \(End_ RG\) of torsion-free, torsion, and mixed R- modules G where R is a commutative ring with identity. \(End_ RG\) is considered as equipped with the finite topology. A central role plays the ideal Ines G of all ”inessential” endomorphisms of G. Conditions are given for a topological R-algebra A which guarantee the existence of a torsion-free, torsion, or mixed R-module G with the property that A can be topologically embedded in \(End_ RG\) in such a way that \(End_ RG=A\oplus Ines G\). The cardinality of the class of non-isomorphic G with this property is considered as well as the existence of ”maximal rigid systems”. Applications are given. Of special interest is an appendix outlining ”The Black Box”.
Reviewer: J.Hausen
MSC:
20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |
16S50 | Endomorphism rings; matrix rings |
13C05 | Structure, classification theorems for modules and ideals in commutative rings |
03E05 | Other combinatorial set theory |
03C60 | Model-theoretic algebra |