The role of the Jacobson radical in isomorphism theorems. (English) Zbl 1264.20054
Strüngmann, Lutz (ed.) et al., Groups and model theory. In honor of Rüdiger Göbel’s 70th birthday. Proceedings of the conference, conference center “Die Wolfsburg”, Mühlheim an der Ruhr, Germany, May 30–June 3, 2011. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-6923-9/pbk; 978-0-8218-9098-1/ebook). Contemporary Mathematics 576, 77-88 (2012).
Let \(R\) be a discrete valuation domain and let \(M\) and \(N\) be mixed \(R\)-modules. Let \(J_M\) and \(J_N\) be the Jacobson radicals of the respective endomorphism rings \(\mathrm{End}(M)\) and \(\mathrm{End}(N)\). If \(J_M\) and \(J_N\) are isomorphic rings, what can be said about the structures of \(M\) and \(N\)?
The author proves that if the torsion module \(T(M)\) of \(M\) has an unbounded basic submodule, then any \(R\)-isomorphism of \(J_M\) onto \(J_N\) induces an isomorphism of \(T(M)\) onto \(T(N)\). Furthermore, the same conclusion holds if \(R\) is complete, \(M\) and \(N\) have torsion-free rank 1, and \(T(M)\) is totally projective and unbounded. Finally, if \(R\) is complete and \(M\) and \(N\) are reduced Warfield modules of finite torsion-free rank, then \(M\cong N\).
For the entire collection see [Zbl 1248.20002].
The author proves that if the torsion module \(T(M)\) of \(M\) has an unbounded basic submodule, then any \(R\)-isomorphism of \(J_M\) onto \(J_N\) induces an isomorphism of \(T(M)\) onto \(T(N)\). Furthermore, the same conclusion holds if \(R\) is complete, \(M\) and \(N\) have torsion-free rank 1, and \(T(M)\) is totally projective and unbounded. Finally, if \(R\) is complete and \(M\) and \(N\) are reduced Warfield modules of finite torsion-free rank, then \(M\cong N\).
For the entire collection see [Zbl 1248.20002].
Reviewer: Phillip Schultz (Perth)
MSC:
| 20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |
| 13C13 | Other special types of modules and ideals in commutative rings |
| 16N20 | Jacobson radical, quasimultiplication |
| 20K21 | Mixed groups |
| 16S50 | Endomorphism rings; matrix rings |
| 13C05 | Structure, classification theorems for modules and ideals in commutative rings |
