Nilpotent Subsets of Graded Algebras with Color Involution

In this article, we examine group-graded algebras R with color involution. If S is the symmetric elements and K the skew symmetric elements, it is shown that if S ⁿ  = 0 then R ³ⁿ  = 0, whereas if K ⁿ  = 0 then (RR, R]_ε R) ⁿ  = 0, where RR, R]_ε R is the ε-commutator ideal corresponding to a bicharacter ε. It is then shown that the bounds found for the degrees of nilpotence of R and RR, R]_ε R are best possible. We also examine the situation where R is graded by a finite group and only the identity component of either S or K is assumed to be nilpotent. We conclude by looking at some questions related to the Nagata-Higman theorem.