An extension of Zassenhaus' theorem on endomorphism rings

• Autorzy: Manfred Dugas , Rüdiger Göbel
• Języki publikacji: EN

Abstrakty

EN

Let R be a ring with identity such that R⁺, the additive group of R, is torsion-free. If there is some R-module M such that $R ⊆ M ⊆ ℚR (= ℚ ⊗_{ℤ} R)$ and $End_{ℤ}(M) = R$, we call R a Zassenhaus ring. Hans Zassenhaus showed in 1967 that whenever R⁺ is free of finite rank, then R is a Zassenhaus ring. We will show that if R⁺ is free of countable rank and each element of R is algebraic over ℚ, then R is a Zassenhaus ring. We will give an example showing that this restriction on R is needed. Moreover, we will show that a ring due to A. L. S. Corner, answering Kaplansky's test problems in the negative for torsion-free abelian groups, is a Zassenhaus ring.

Kategorie tematyczne

• 16S60: Rings of functions, subdirect products, sheaves of rings
• 16W20: Automorphisms and endomorphisms
• 20K30: Automorphisms, homomorphisms, endomorphisms, etc.
• 20K20: Torsion-free groups, infinite rank

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2007
• Tom: 194
• Numer: 3
• Strony: 239-251

Daty

wydano

2007

Twórcy

autor

Manfred Dugas

• Department of Mathematics, Baylor University, Waco, TX 76798, U.S.A.

autor

Rüdiger Göbel

• Fachbereich Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany

Identyfikatory

DOI

10.4064/fm194-3-2