{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T13:50:38Z","timestamp":1770817838865,"version":"3.50.1"},"reference-count":10,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","funder":[{"name":"Spanish Government","award":["PID2020-117281GB-I00"],"award-info":[{"award-number":["PID2020-117281GB-I00"]}]},{"name":"Spanish Government","award":["PID2019-107444GAI00"],"award-info":[{"award-number":["PID2019-107444GAI00"]}]},{"DOI":"10.13039\/501100003086","name":"Basque Government","doi-asserted-by":"crossref","award":["IT1483-22"],"award-info":[{"award-number":["IT1483-22"]}],"id":[{"id":"10.13039\/501100003086","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:p>Let [Formula: see text] be the probability that two random elements of a finite ring R commute and [Formula: see text] the probability that the product of two random elements in R is zero. We show that if [Formula: see text], then there exists a Lie-ideal D in the Lie-ring [Formula: see text] with [Formula: see text]-bounded index and with [Formula: see text] of [Formula: see text]-bounded order. If [Formula: see text], then there exists an ideal D in R with [Formula: see text]-bounded index and [Formula: see text] of [Formula: see text]-bounded order. These results are analogous to the well-known theorem of Neumann on the commuting probability in finite groups.<\/jats:p>","DOI":"10.1142\/s0218196724500061","type":"journal-article","created":{"date-parts":[[2024,1,31]],"date-time":"2024-01-31T05:01:45Z","timestamp":1706677305000},"page":"201-206","source":"Crossref","is-referenced-by-count":2,"title":["Commuting and product-zero probability in finite rings"],"prefix":"10.1142","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4976-5675","authenticated-orcid":false,"given":"Pavel","family":"Shumyatsky","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Brasilia, Brasilia DF, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1299-3631","authenticated-orcid":false,"given":"Matteo","family":"Vannacci","sequence":"additional","affiliation":[{"name":"Matematika Saila, UPV-EHU, Bilbao, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2024,2,20]]},"reference":[{"issue":"2","key":"S0218196724500061BIB001","doi-asserted-by":"crossref","first-page":"189","DOI":"10.13069\/jacodesmath.284962","volume":"4","author":"Buckley S. M.","year":"2017","journal-title":"J. Algebra Comb. Discrete Struct. Appl."},{"key":"S0218196724500061BIB002","doi-asserted-by":"publisher","DOI":"10.1017\/S0004972721001246"},{"key":"S0218196724500061BIB003","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/bdv050"},{"key":"S0218196724500061BIB004","doi-asserted-by":"publisher","DOI":"10.1142\/S0219498818500548"},{"key":"S0218196724500061BIB005","doi-asserted-by":"publisher","DOI":"10.1142\/S0219498819500762"},{"key":"S0218196724500061BIB006","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1973.11993437"},{"key":"S0218196724500061BIB007","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1976.11994032"},{"key":"S0218196724500061BIB008","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-29.2.236"},{"key":"S0218196724500061BIB009","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/21.5.456"},{"key":"S0218196724500061BIB010","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1957.0007"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196724500061","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,9]],"date-time":"2024-11-09T16:26:46Z","timestamp":1731169606000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0218196724500061"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,20]]},"references-count":10,"journal-issue":{"issue":"02","published-print":{"date-parts":[[2024,3]]}},"alternative-id":["10.1142\/S0218196724500061"],"URL":"https:\/\/doi.org\/10.1142\/s0218196724500061","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,20]]}}}