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Research articles

General method for defining finite non-commutative associative algebras of dimension $m>1$

A. A. Moldovyan

St. Petersburg Institute for Informatics and Automation of Russian Academy of Sciences, 14-th line 39, 199178, St. Petersburg, Russia

Abstract: General method for defining non-commutative finite associative algebras of arbitrary dimension $m\ge2$ is discussed. General formulas describing local unit elements (the right-, left-, and bi-side ones), square roots of zero and zero divisors are derived. For arbitrary value $m$ the single bi-side unit corresponds to every element of the algebra, except the square roots from zero. Various modifications of the multiplication operation can be assigned using different sets of the values of structural coefficients. It is proved that all of the modifications are mutually associative.

Keywords and phrases: finite associative algebra, non-commutative algebra, structural coefficient, mutual associativity, local unit.

MSC: 94A60, 16Z05, 14G50, 11T71, 16S50

Received: 02.05.2018

Language: English