On Graded Associative Algebras

Consider U an associative algebra of arbitrary dimension and over an arbitrary base field K , graded by means of an abelian group G. We show that U is of the form U = U + ∑ j I j with U a lmear subspace of U 1 and any Ij a well described graded ideal of U , satisfymg I j I k = 0 if j ≠ k . Under certam conditions, the simplicity of U is characterized and it is shown that U is the direct sum of the family of its minimal graded ideals, each one being a simple graded associative algebra.