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Additive subgroups generated by noncommutative polynomials

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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, \(S_k(R)\) is completely determined for any standard polynomial \(S_k\) when R is generated by \(S_k(R)\) as an ideal. Motivated by Brešar’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.

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Acknowledgements

The work was supported in part by the Ministry of Science and Technology of Taiwan (MOST 109-2115-M-002-014).

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  1. Department of Mathematics, National Taiwan University, 106, Taipei, Taiwan

    Tsiu-Kwen Lee

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  1. Tsiu-Kwen Lee
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Correspondence to Tsiu-Kwen Lee.

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Communicated by John S. Wilson.

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Lee, TK. Additive subgroups generated by noncommutative polynomials. Monatsh Math 199, 149–165 (2022). https://doi.org/10.1007/s00605-021-01644-3

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  • DOI: https://doi.org/10.1007/s00605-021-01644-3

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