In the paper,we study algebras having n bilinearmultiplication operations [InlineMediaObject not available: see fulltext.]: A× A → A, s = 1, ..., n, such that ( a[InlineMediaObject not available: see fulltext.] b) [InlineMediaObject not available: see fulltext.] c = a[InlineMediaObject not available: see fulltext.] ( b[InlineMediaObject not available: see fulltext.] 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 [InlineMediaObject not available: see fulltext.]. 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. [ABSTRACT FROM AUTHOR]