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An extension of Zassenhaus' theorem on endomorphism rings

Manfred Dugas, Rüdiger Göbel (2007)

Fundamenta Mathematicae

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 ⊆ ℚ R ( = ℚ ⊗ ℤ R ) and E n d 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....

A-Rings

Manfred Dugas, Shalom Feigelstock (2003)

Colloquium Mathematicae

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...

Cayley-Hamilton Theorem for Matrices over an Arbitrary Ring

Szigeti, Jeno (2006)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.For an n×n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients: (λ0I+C0)+A(λ1I+C1)+… +An-1(λn-1I+Cn-1)+An (n!I+Cn) = 0, where λ0+λ1x+…+λn-1 xn-1+n!xn is the right characteristic polynomial of A in R[x], I ∈ Mn(R) is the identity matrix and the entries of the n×n matrices Ci, 0 ≤ i ≤ n are in [R,R]. If R is commutative, then C0 = C1 = … = Cn-1 = Cn = 0 and our...

Construction of Auslander-Gorenstein local rings as Frobenius extensions

Mitsuo Hoshino, Noritsugu Kameyama, Hirotaka Koga (2015)

Colloquium Mathematicae

Starting from an arbitrary ring R we provide a systematic construction of ℤ/nℤ-graded rings A which are Frobenius extensions of R, and show that under mild assumptions, A is an Auslander-Gorenstein local ring if and only if so is R.

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