On the identical relations of associative and Lie algebras equipped with an action

Let R be a unitary associative algebra over a field. We call an algebra A a generalized R-algebra when A is endowed with an R-module action with the property that, for each r is an element of R, there exists finitely many elements r(+) = (r(1)(+), r(2)(+)) is an element of R-2 and r(-) = (r(1)(-) , r(2)(-)) is an element of R-2 such that, for all a(1), a(2) is an element of A r center dot (a(1)a(2)) = Sigma(r+) (r(1)(+) center dot a(1)) (r(2)(+) center dot a(1)) + Sigma(r-) (r(2)(-) center dot a(2)) + (r(1)(-) center dot a(1)) Suppose an associative generalized R-algebra A satisfies an identical relation of the form x(1) center dot center dot center dot Sigma 1 not equal sigma is an element of S-d Sigma(r) (r1 center dot x(sigma(1))) center dot center dot center dot (r(d) center dot x(sigma(d))) equivalent to 0, where S-d denotes the symmetric group of degree d and the inner sum runs over finitely many r = (r(1), . . . , r(d)) is an element of R-d.We prove: if the algebra of endomorphisms on A defined by the action of R is m-dimensional, then A satisfies a classical polynomial identity of degree bounded by an explicit function of d and m only. We also prove the analogous result holds when A is a Lie algebra, thus extending a collection of results in associative and Lie PI-theory. (c) 2022 Elsevier Inc. All rights reserved.