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A Basis for Z-Graded Identities of Matrices over Infinite Fields

Azevedo, Sergio (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 16R10, 16R20, 16R50The algebra Mn(K) of the matrices n × n over a field K can be regarded as a Z-graded algebra. In this paper, it is proved that if K is an infinite field, all the Z-graded polynomial identities of Mn(K) follow from the identities: x = 0, |α(x)| ≥ n, xy = yx, α(x) = α(y) = 0, xyz = zyx, α(x) = −α(y) = α(z ), where α is the degree of the corresponding variable. This is a generalization of a result of Vasilovsky about the Z-graded identities...

A canonical directly infinite ring

Mario Petrich, Pedro V. Silva (2001)

Czechoslovak Mathematical Journal

Let ℕ be the set of nonnegative integers and ℤ the ring of integers. Let ℬ be the ring of N × N matrices over ℤ generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of ℬ yields that the subrings generated by them coincide. This subring is the sum of the ideal ℱ consisting of...

A class of quasitilted rings that are not tilted

Riccardo Colpi, Kent R. Fuller, Enrico Gregorio (2006)

Colloquium Mathematicae

Based on the work of D. Happel, I. Reiten and S. Smalø on quasitilted artin algebras, the first two authors recently introduced the notion of quasitilted rings. Various authors have presented examples of quasitilted artin algebras that are not tilted. Here we present a class of right quasitilted rings that not right tilted, and we show that they satisfy a condition that would force a quasitilted artin algebra to be tilted.

A generalization of Mathieu subspaces to modules of associative algebras

Wenhua Zhao (2010)

Open Mathematics

We first propose a generalization of the notion of Mathieu subspaces of associative algebras 𝒜 , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to 𝒜 -modules ℳ . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable...

A note on Skolem-Noether algebras

Juncheol Han, Tsiu-Kwen Lee, Sangwon Park (2021)

Czechoslovak Mathematical Journal

The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let K be a unital commutative ring, not necessarily a field. Given a unital K -algebra S , where K is contained in the center of S , n ∈ ℕ , the goal of this paper is to study the question: when can a homomorphism φ : M n ( K ) → M n ( S ) be extended to an inner automorphism of M n ( S ) ? As an application of main results presented in the paper, it is proved that if S is a semilocal...

A Survey of Rings Generated by Units

Ashish K. Srivastava (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

This article presents a brief survey of the work done on rings generated by their units.

Actions of parabolic subgroups in GL_n on unipotent normal subgroups and quasi-hereditary algebras

Thomas Brüstle, Lutz Hille (2000)

Colloquium Mathematicae

Let R be a parabolic subgroup in G L n . It acts on its unipotent radical R u and on any unipotent normal subgroup U via conjugation. Let Λ be the path algebra k t of a directed Dynkin quiver of type with t vertices and B a subbimodule of the radical of Λ viewed as a Λ-bimodule. Each parabolic subgroup R is the group of automorphisms of an algebra Λ(d), which is Morita equivalent to Λ. The action of R on U can be described using matrices over the bimodule B. The advantage of this description is that each...

AE-rings

Manfred Dugas, Shalom Feigelstock (2004)

Rendiconti del Seminario Matematico della Università di Padova

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....

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