On linear equations in some non-commutative algebras

Jonas Månsson

On linear equations in some non-commutative algebras

Research output: Contribution to conference › Paper, not in proceeding

Abstract

The problem of solving linear equations in a non-commutative algebra is in general a highly non-trivial matter. Even in the case of finitely presented algebras, there is no general algorithms for solving seemingly simple equations of the type a X = X b for some elements a and b.

In this paper we will demonstrate a method by which it is possible to find all the solutions to linear equations in certain factor algebras of the noncommutative polynomial ring. The commutative case reduces to computing syzygy modules, which is treated in Adams [1]. Here we will consider algebras the center of which is sufficiently large, in the sense that the former can be considered a Noetherian module over a subalgebra of its center. We will show that with the aid of Groebner
basis technique, the problem of finding the solutions in the non-commutative setting can be reduced to computing a syzygy module.

Original language	English
Publication status	Published - 1999
Event	FLoC'99 Workshop, Gröbner Bases and Rewriting Techniques - Trento, Italy Duration: 1999 Jun 30 → 1999 Jul 1

Conference

Conference	FLoC'99 Workshop, Gröbner Bases and Rewriting Techniques
Country/Territory	Italy
City	Trento
Period	1999/06/30 → 1999/07/01

Bibliographical note

The FLoC'99 Workshop was held preliminary to the 10th International Conference
Rewriting Techniques and Applications (RTA-99), Trento, Italy, July 2-4, 1999.

More information about the workshop is found at
http://www.math.unl.edu/~shermiller2/conf/floc99workshop8

Subject classification (UKÄ)

Mathematical Sciences

Cite this

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title = "On linear equations in some non-commutative algebras",

abstract = "The problem of solving linear equations in a non-commutative algebra is in general a highly non-trivial matter. Even in the case of finitely presented algebras, there is no general algorithms for solving seemingly simple equations of the type a X = X b for some elements a and b. In this paper we will demonstrate a method by which it is possible to find all the solutions to linear equations in certain factor algebras of the noncommutative polynomial ring. The commutative case reduces to computing syzygy modules, which is treated in Adams [1]. Here we will consider algebras the center of which is sufficiently large, in the sense that the former can be considered a Noetherian module over a subalgebra of its center. We will show that with the aid of Groebner basis technique, the problem of finding the solutions in the non-commutative setting can be reduced to computing a syzygy module.",

author = "Jonas M{\aa}nsson",

note = "The FLoC'99 Workshop was held preliminary to the 10th International Conference Rewriting Techniques and Applications (RTA-99), Trento, Italy, July 2-4, 1999. More information about the workshop is found at http://www.math.unl.edu/~shermiller2/conf/floc99workshop8; FLoC'99 Workshop, Gr{\"o}bner Bases and Rewriting Techniques ; Conference date: 30-06-1999 Through 01-07-1999",

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language = "English",

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T1 - On linear equations in some non-commutative algebras

AU - Månsson, Jonas

N1 - The FLoC'99 Workshop was held preliminary to the 10th International Conference Rewriting Techniques and Applications (RTA-99), Trento, Italy, July 2-4, 1999. More information about the workshop is found at http://www.math.unl.edu/~shermiller2/conf/floc99workshop8

PY - 1999

Y1 - 1999

N2 - The problem of solving linear equations in a non-commutative algebra is in general a highly non-trivial matter. Even in the case of finitely presented algebras, there is no general algorithms for solving seemingly simple equations of the type a X = X b for some elements a and b. In this paper we will demonstrate a method by which it is possible to find all the solutions to linear equations in certain factor algebras of the noncommutative polynomial ring. The commutative case reduces to computing syzygy modules, which is treated in Adams [1]. Here we will consider algebras the center of which is sufficiently large, in the sense that the former can be considered a Noetherian module over a subalgebra of its center. We will show that with the aid of Groebner basis technique, the problem of finding the solutions in the non-commutative setting can be reduced to computing a syzygy module.

AB - The problem of solving linear equations in a non-commutative algebra is in general a highly non-trivial matter. Even in the case of finitely presented algebras, there is no general algorithms for solving seemingly simple equations of the type a X = X b for some elements a and b. In this paper we will demonstrate a method by which it is possible to find all the solutions to linear equations in certain factor algebras of the noncommutative polynomial ring. The commutative case reduces to computing syzygy modules, which is treated in Adams [1]. Here we will consider algebras the center of which is sufficiently large, in the sense that the former can be considered a Noetherian module over a subalgebra of its center. We will show that with the aid of Groebner basis technique, the problem of finding the solutions in the non-commutative setting can be reduced to computing a syzygy module.

M3 - Paper, not in proceeding

T2 - FLoC'99 Workshop, Gröbner Bases and Rewriting Techniques

Y2 - 30 June 1999 through 1 July 1999

ER -

On linear equations in some non-commutative algebras

Abstract

Conference

Bibliographical note

Subject classification (UKÄ)

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Cite this