Geodesic
Associative $n$-Tuple Algebras
Matematičeskie zametki, Tome 96 (2014) no. 1, pp. 36-50
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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. 
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N. A. Koreshkov. Associative $n$-Tuple Algebras. Matematičeskie zametki, Tome 96 (2014) no. 1, pp. 36-50. http://geodesic.mathdoc.fr/item/MZM_2014_96_1_a3/

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