{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T11:28:55Z","timestamp":1771673335423,"version":"3.50.1"},"reference-count":17,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,30]],"date-time":"2024-12-30T00:00:00Z","timestamp":1735516800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at King Khalid University (KKU), Abha, Saudi Arabia","award":["RGP. 2\/293\/45"],"award-info":[{"award-number":["RGP. 2\/293\/45"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The current article focuses on studying the behavior of a ring \u211c\/\u03a0 when \u211c admits generalized derivations \u03a8 and \u03a9 with associated derivations \u03d5 and \u03b4, respectively. These derivations satisfy specific differential identities involving \u03a0, where \u03a0 is a prime ideal of an arbitrary ring \u211c, not necessarily prime or semiprime. Furthermore, we explore some consequences of our findings. To emphasize the necessity of the primeness of \u03a0 in the hypotheses of our various theorems, we provide a list of examples.<\/jats:p>","DOI":"10.3390\/axioms14010015","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T07:00:37Z","timestamp":1735628437000},"page":"15","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Factor Rings with Algebraic Identities via Generalized Derivations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8516-0868","authenticated-orcid":false,"given":"Ali Yahya","family":"Hummdi","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 61471, Saudi Arabia"}]},{"given":"Zakia Z.","family":"Al-Amery","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aden University, Aden 5243, Yemen"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5334-6169","authenticated-orcid":false,"given":"Radwan M.","family":"Al-omary","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ibb University, Ibb 70270, Yemen"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1093","DOI":"10.1090\/S0002-9939-1957-0095863-0","article-title":"Derivations in prime rings","volume":"8","author":"Posner","year":"1957","journal-title":"Proc. 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