Associative n-Tuple algebras

In the paper,we study algebras having n bilinearmultiplication operations : A × A → A , s = 1, …, n , such that ( a b ) c = a ( b c ), s, r = 1,..., n, a, b, c ∈ 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 . 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.