On linear equations in some non-commutative algebras

Research output: Contribution to conferencePaper, not in proceeding


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 languageEnglish
Publication statusPublished - 1999
EventFLoC'99 Workshop, Gröbner Bases and Rewriting Techniques - Trento, Italy
Duration: 1999 Jun 301999 Jul 1


ConferenceFLoC'99 Workshop, Gröbner Bases and Rewriting Techniques

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

Subject classification (UKÄ)

  • Mathematics

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