LAUSR

Abstract In this paper we generalize the concepts of good and elementary gradings for an associative algebra A with a fixed multiplicative basis B. When the group G considered in the grading is abelian, we equip the set of good G-gradings of A with a structure of abelian group, which is denoted by G ( B , G ) . Moreover, when A admits elementary G-gradings, we show the set E ( B , G ) of all elementary G-gradings of A is a subgroup of G ( B , G ) . In this case, we introduce a cohomology for the pair ( A , B ) and we show that G ( B , G ) / E ( B , G ) is isomorphic to the first cohomology group of ( A , B ) with coefficients in G.