Simple graded commutative algebras

We study the notion of Γ-graded commutative algebra for an arbitrary abelian group Γ. The main examples are the Clifford algebras already treated in H. Albuquerque and S. Majid (2002) [2]. We prove that the Clifford algebras are the only simple finite-dimensional associative graded commutative algebras over R or C. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.