On the structure of graded commutative algebras

Consider A a commutative algebra graded by means of an abelian group G. We show that, under the hypothesis A0=∑g∈ΣAgA-g where Σ={g∈G:Ag≠0}, the algebra A is of the form A=∑jIj with any Ij a well described graded ideal of A, satisfying IjIk=0 if j≠k. Under certain conditions, the simplicity of A is characterized and it is shown that A is the direct sum of the family of its minimal graded ideals, each one being a simple graded commutative algebra.