TSIU-KWEN LEE2022-04-252022-04-25202100269255https://www.scopus.com/inward/record.uri?eid=2-s2.0-85119342081&doi=10.1007%2fs00605-021-01644-3&partnerID=40&md5=3994a0862cf545ed291004d14b65bc17https://scholars.lib.ntu.edu.tw/handle/123456789/606440Let 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.Higher commutatorLie idealMaximal idealNoncommutative polynomialPI-algebraSimple algebraStandard polynomialjournal article10.1007/s00605-021-01644-32-s2.0-85119342081