{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T07:57:17Z","timestamp":1648799837959},"reference-count":17,"publisher":"World Scientific Pub Co Pte Lt","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2017,12]]},"abstract":"<jats:p> Let [Formula: see text] be a field of characteristic [Formula: see text] and let [Formula: see text] be a unital associative [Formula: see text]-algebra. Define a left-normed commutator [Formula: see text] [Formula: see text] recursively by [Formula: see text], [Formula: see text] [Formula: see text]. For [Formula: see text], let [Formula: see text] be the two-sided ideal in [Formula: see text] generated by all commutators [Formula: see text] ([Formula: see text]. Define [Formula: see text]. <\/jats:p><jats:p> Let [Formula: see text] be integers such that [Formula: see text], [Formula: see text]. Let [Formula: see text] be positive integers such that [Formula: see text] of them are odd and [Formula: see text] of them are even. Let [Formula: see text]. The aim of the present note is to show that, for any positive integers [Formula: see text], in general, [Formula: see text]. It is known that if [Formula: see text] (that is, if at least one of [Formula: see text] is even), then [Formula: see text] for each [Formula: see text] so our result cannot be improved if [Formula: see text]. <\/jats:p><jats:p> Let [Formula: see text]. Recently, Dangovski has proved that if [Formula: see text] are any positive integers then, in general, [Formula: see text]. Since [Formula: see text], Dangovski\u2019s result is stronger than ours if [Formula: see text] and is weaker than ours if [Formula: see text]; if [Formula: see text], then [Formula: see text] so both results coincide. It is known that if [Formula: see text] (that is, if all [Formula: see text] are odd) then, for each [Formula: see text], [Formula: see text] so in this case Dangovski\u2019s result cannot be improved. <\/jats:p>","DOI":"10.1142\/s0218196717500473","type":"journal-article","created":{"date-parts":[[2017,9,21]],"date-time":"2017-09-21T22:13:25Z","timestamp":1506032005000},"page":"1027-1040","source":"Crossref","is-referenced-by-count":1,"title":["Products of several commutators in a Lie nilpotent associative algebra"],"prefix":"10.1142","volume":"27","author":[{"given":"Galina","family":"Deryabina","sequence":"first","affiliation":[{"name":"Department of Computational Mathematics and Mathematical Physics (FS-11), Bauman Moscow State Technical University, 2-nd Baumanskaya Street, 5, 105005 Moscow, Russia"}]},{"given":"Alexei","family":"Krasilnikov","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidade de Bras\u00edlia, 70910-900 Bras\u00edlia, DF, Brasil"}]}],"member":"219","published-online":{"date-parts":[[2017,12,28]]},"reference":[{"key":"S0218196717500473BIB001","doi-asserted-by":"publisher","DOI":"10.1142\/S0219498816501875"},{"key":"S0218196717500473BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2012.10.001"},{"key":"S0218196717500473BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2012.07.052"},{"key":"S0218196717500473BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-010-0074-1"},{"key":"S0218196717500473BIB007","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2015.01.009"},{"key":"S0218196717500473BIB008","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2016.08.031"},{"key":"S0218196717500473BIB009","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2008.09.042"},{"key":"S0218196717500473BIB010","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2007.v14.n5.a7"},{"key":"S0218196717500473BIB011","doi-asserted-by":"publisher","DOI":"10.1070\/RM2007v062n01ABEH004383"},{"key":"S0218196717500473BIB012","doi-asserted-by":"publisher","DOI":"10.1070\/SM2015v206n11ABEH004506"},{"key":"S0218196717500473BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/s10958-011-0515-3"},{"key":"S0218196717500473BIB014","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196795000203"},{"key":"S0218196717500473BIB015","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(83)90217-X"},{"key":"S0218196717500473BIB016","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2013.06.021"},{"key":"S0218196717500473BIB017","first-page":"1432","volume":"6","author":"Latyshev V. N.","year":"1965","journal-title":"Sibirsk. Mat. Zh."},{"key":"S0218196717500473BIB018","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(85)90085-4"},{"key":"S0218196717500473BIB019","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(90)90056-N"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196717500473","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T08:25:34Z","timestamp":1565079934000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196717500473"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12]]},"references-count":17,"journal-issue":{"issue":"08","published-online":{"date-parts":[[2017,12,28]]},"published-print":{"date-parts":[[2017,12]]}},"alternative-id":["10.1142\/S0218196717500473"],"URL":"https:\/\/doi.org\/10.1142\/s0218196717500473","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,12]]}}}