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An extension of Zassenhaus' theorem on endomorphism rings

100%

Dugas M., Göbel R.

Fundamenta Mathematicae

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2007

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tom 194

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nr 3

239-251

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.

2

A-Rings

100%

Dugas M., Feigelstock S.

Colloquium Mathematicum

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2003

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tom 96

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nr 2

277-292

EN

A ring R is called an E-ring if every endomorphism of R⁺, the additive group of R, is multiplication on the left by an element of R. This is a well known notion in the theory of abelian groups. We want to change the "E" as in endomorphisms to an "A" as in automorphisms: We define a ring to be an A-ring if every automorphism of R⁺ is multiplication on the left by some element of R. We show that many torsion-free finite rank (tffr) A-rings are actually E-rings. While we have an example of a mixed A-ring that is not an E-ring, it is still open if there are any tffr A-rings that are not E-rings. We will employ the Strong Black Box [5] to construct large integral domains that are A-rings but not E-rings.

3

Quotients of reflexive modules

50%

Dugas M., Göbel R.

Fundamenta Mathematicae

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1981

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tom 114

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nr 1

17-28

Ograniczanie wyników

2 Fundamenta Mathematicae

1 Colloquium Mathematicum

3 Dugas M.

2 Göbel R.

1 Feigelstock S.

1 2007

1 2003

1 1981

An extension of Zassenhaus' theorem on endomorphism rings

A-Rings

Quotients of reflexive modules