Symmetric identities in graded algebras

Let P k be the symmetric polynomial of degree k i.e., the full linearization of the polynomial x k . Let G be a cancellation semigroup with 1 and R a G-graded ring with finite support of order n. We prove that if R 1 satisfies $ P_k \equiv 0 $ then R satisfies $ P_{kn} \equiv 0 $ .

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σg∈GRg be any associative algebra over a field such that the subspacesRg satisfyRgRh⊆Rgh. We prove that ifR1 satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withRH satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.

Identities of sums of commutative subalgebras

SiaR un'algebra associativa tale cheR=A+B conA, B sottoalgebre commutative. Si dimostra cheR soddisfa l'identita polinomiale [[x,y],[z,t]]≡0 e che, seV e la varieta determinata da questa identita,V e la piu piccola varieta contenente tutte le algebre somma di sottoalgebre commutative. Si determina inoltre la struttura delle algebre libere diV.

G-identities on associative algebras