On Graded Associative Algebras *

Consider UU an associative algebra of arbitrary dimension and over an arbitrary base field K,K, graded by means of an abelian group G  . We show that UU is of the form U=U+∑jIj with U   a lmear subspace of U1U1 and any Ij   a well described graded ideal of UU, satisfymg IjIk=0IjIk=0 if j≠kj≠k. Under certam conditions, the simplicity of UU is characterized and it is shown that UU is the direct sum of the family of its minimal graded ideals, each one being a simple graded associative algebra.