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5 papers
Associative $n$-Tuple Algebras
N. A. Koreshkov Kazan (Volga Region) Federal University
Abstract:
In the paper, we study algebras having
$n$ bilinear multiplication operations
$\boxed{s}\colon A\times A\to A$,
$s=1,\dots,n$, such that $(a\mathbin{\boxed{s}}b)\mathbin{\boxed{r}}c= a\mathbin{\boxed{s}}(b\mathbin{\boxed{r}}c)$,
$s,r=1,\dots,n$,
$a,b,c\in A$. The radical of such an algebra is defined as the intersection of the annihilators of irreducible
$A$-modules, and it is proved that the radical coincides with the intersection of the maximal right ideals each of which is
$s$-regular for some operation
$\boxed{s}$ . This implies that the quotient algebra by the radical is semisimple. If an
$n$-tuple algebra is Artinian, then the radical is nilpotent, and the semisimple Artinian
$n$-tuple algebra is the direct sum of two-sided ideals each of which is a simple algebra. Moreover, in terms of sandwich algebras, we describe a finite-dimensional
$n$-tuple algebra
$A$, over an algebraically closed field, which is a simple
$A$-module.
Keywords:
$n$-tuple algebra, radical, semisimple algebra, Artinian algebra, sandwich algebra, commutator algebra.
UDC:
512.554 Received: 10.07.2013
DOI:
10.4213/mzm8889
Fulltext:
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