Additive subgroups generated by noncommutative polynomials
Journal
Monatshefte fur Mathematik
Date Issued
2021
Author(s)
Abstract
Let R be an algebra. Given a noncommutative polynomial f, let f(R) stand for the additive subgroup of R generated by the image of f. For a unital or an affine algebra R, Sk(R) is completely determined for any standard polynomial Sk when R is generated by Sk(R) as an ideal. Motivated by Bre?ar’s paper [Adv. Math. 374 (2020), 107346, 21 pp] and Robert’s paper [J. Oper. Theory 75 (2016), 387–408], under certain conditions we also prove that f(R) is equal to either [R,?R] or the whole ring R. We obtain these results by studying the structure of Lie ideals L of a ring R whenever R is generated by [R,?L] as an ideal. ? 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.
Subjects
Higher commutator
Lie ideal
Maximal ideal
Noncommutative polynomial
PI-algebra
Simple algebra
Standard polynomial
Type
journal article