Zq-graded identities and central polynomials of the Grassmann algebra

Let F be an infinite field of characteristic p different from 2 and let E be the Grassmann algebra generated by an infinite dimensional vector space L over F. In this paper we provide, for any odd prime q, a finite basis for the T-q-ideal of the Z(q) -graded polynomial identities for E and a basis for the T-q space of graded central polynomials for E, for any Z(q)-grading on E such that L is homogeneous in the grading. Moreover, we prove that the set of all graded central polynomials of E is not finitely generated as a T-q-space, if p > 2. In the nonhomogeneous case such bases are also described when at least one non-neutral component has infinite many homogeneous elements of the basis of L in the respective grading. (C) 2020 Elsevier Inc. All rights reserved.