A note on Z-gradings on the Grassmann algebra and elementary number theory

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 E =E-(r1,r2,r3)((v1,v2,v3)) in which each element of a basis of L has Z-degree r(1), r(2), or r(3). We provide a criterion for the support of this structure to coincide with a subgroup of the group and we describe the graded identities for the cor responding 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 Brandao A, Fidelis C, Guimaraes A. Z-gradings of full support on the Grassmann algebra.