A note on $\mathbb{Z}$-gradings on the Grassmann algebra and Elementary Number Theory

Abstract

Let E be the Grassmann algebra of an infinite dimensional vector space L over a field of characteristic zero. In this paper, we study the Z-gradings on E having the form, in which each element of a basis of L has Z-degree r1, r2, or r3. We provide a criterion for the support of this structure to coincide with a subgroup of the group Z, and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in [11].