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Classification of finite rings: theory and algorithm

Mahmood Behboodi, Reza Beyranvand, Amir Hashemi, Hossein Khabazian (2014)

Czechoslovak Mathematical Journal

An interesting topic in the ring theory is the classification of finite rings. Although rings of certain orders have already been classified, a full description of all rings of a given order remains unknown. The purpose of this paper is to classify all finite rings (up to isomorphism) of a given order. In doing so, we introduce a new concept of quasi basis for certain type of modules, which is a useful computational tool for dealing with finite rings. Then, using this concept, we give structure...

Degree estimate for subalgebras generated by two elements

Leonid Makar-Limanov, Jie-Tai Yu (2008)

Journal of the European Mathematical Society

We develop a new combinatorial method to deal with a degree estimate for subalgebras generated by two elements in different environments. We obtain a lower bound for the degree of the elements in two-generated subalgebras of a free associative algebra over a field of zero characteristic. We also reproduce a somewhat refined degree estimate of Shestakov and Umirbaev for the polynomial algebra, which plays an essential role in the recent celebrated solution of the Nagata conjecture and the strong...

Free associative algebras, noncommutative Gröbner bases, and universal associative envelopes for nonassociative structures

Murray R. Bremner (2014)

Commentationes Mathematicae Universitatis Carolinae

First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.

On computation of minimal free resolutions over solvable polynomial algebras

Huishi Li (2015)

Commentationes Mathematicae Universitatis Carolinae

Let A = K [ a 1 , ... , a n ] be a (noncommutative) solvable polynomial algebra over a field K in the sense of A. Kandri-Rody and V. Weispfenning [Non-commutative Gröbner bases in algebras of solvable type, J. Symbolic Comput. 9 (1990), 1–26]. This paper presents a comprehensive study on the computation of minimal free resolutions of modules over A in the following two cases: (1) A = p A p is an -graded algebra with the degree-0 homogeneous part A 0 = K ; (2) A is an -filtered algebra with the filtration { F p A } p determined by a positive-degree...

Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions

Murray R. Bremner, Sara Madariaga, Luiz A. Peresi (2016)

Commentationes Mathematicae Universitatis Carolinae

This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra 𝔽 S n of the symmetric group S n over a field 𝔽 of characteristic 0 (or p > n ). The goal is to obtain a constructive version of the isomorphism ψ : λ M d λ ( 𝔽 ) 𝔽 S n where λ is a partition of n and d λ counts the standard tableaux of shape λ ....

The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras

Piotr Dowbor, Andrzej Mróz (2008)

Colloquium Mathematicae

Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector m ( M ) = ( m x ) x X X such that M x X X x m x is studied. A precise formula for d i m k H o m Λ ( M , X ) , for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors in Colloq....

The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach

Piotr Dowbor, Andrzej Mróz (2007)

Colloquium Mathematicae

Given a module M over an algebra Λ and a complete set of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector m ( M ) = ( m X ) X such that M X X m X is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type ̃ p , q . In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).

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