Associative $n$-Tuple Algebras

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.