On Lie Nilpotent Rings and Cohen's Theorem
- 21 July 2015
- journal article
- research article
- Published by Taylor & Francis Ltd in Communications in Algebra
- Vol. 43 (11), 4783-4796
- https://doi.org/10.1080/00927872.2014.952735
Abstract
We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L + rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Z(n)(R), n >= 1, which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring < Z(n)(R) boolean OR C > of R generated by the subset Z(n)(R) boolean OR C of R is also Lie nilpotent of index n.Keywords
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