$n$-tuple algebras of associative type

We study properties of $n$-tuple algebras of associative type. We show that the nilpotency of an $n$-tuple algebras of associative type is determined by the nilpotency of each element. In addition, we characterize the nilpotency of an $n$-tuple algebra of associative type in terms of the trace function. In the final part of the paper, we show that a homogeneously semisimple $n$-tuple algebra of associative type is the direct sum of two-sided ideals each of which is a homogeneously simple $n$-tuple algebra of associative type.