Modules and ideals of algebras of associative type

In this paper we study some properties of associative-type algebras introduced in previous papers of the author. We show that a finite-dimensional algebra of associative type over a field of zero characteristic is homogeneously semisimple, if and only if a certain form defined by the trace form is nonsingular. We prove the total reducedness of modulus over semisimple algebras in a certain subclass of associative-type algebras. We also prove that any left homogeneous ideal of a semisimple algebra of associative type is generated by a homogeneous idempotent.