A characterization of the natural grading of the Grassmann algebra and its non-homogeneous [formula omitted]-gradings.

Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we will provide a condition for a Z 2 -grading on E to behave like the natural Z 2 -grading E c a n. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z 2 -grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of E c a n by means of its Z 2 -graded polynomial identities. Furthermore we construct a Z 2 -grading on E that gives a negative answer to the conjecture. [ABSTRACT FROM AUTHOR]