{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T15:54:23Z","timestamp":1774626863749,"version":"3.50.1"},"reference-count":19,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>Let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>be a 2-torsion free ring and let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><\/mml:math>be a noncentral Lie ideal of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>, and let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>F<\/mml:mi><mml:mo>:<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mi>R<\/mml:mi><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>G<\/mml:mi><mml:mo>:<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo>\u2192<\/mml:mo><mml:mi>R<\/mml:mi><\/mml:math>be two generalized derivations of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>. We will analyse the structure of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>in the following cases: (a)<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>is prime and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mi>F<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>for all<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>u<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>L<\/mml:mi><\/mml:math>and fixed positive integers<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mi>m<\/mml:mi><mml:mo>\u2260<\/mml:mo><mml:mi>n<\/mml:mi><\/mml:math>; (b)<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>is prime and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mi>F<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mrow><mml:mi>v<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>r<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mi>u<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>for all<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>L<\/mml:mi><\/mml:math>and fixed integers<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mi>m<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>q<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>s<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math>; (c)<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>is semiprime and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mi>F<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>u<\/mml:mi><mml:mi>v<\/mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mi>u<\/mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>for all<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mo stretchy=\"false\">[<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>R<\/mml:mi><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:math>and fixed integer<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\"><mml:mi>n<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math>; and (d)<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M20\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>is semiprime and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M21\"><mml:mi>F<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>u<\/mml:mi><mml:mi>v<\/mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><mml:mo>=<\/mml:mo><mml:mi>G<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>v<\/mml:mi><mml:mi>u<\/mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>for all<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M22\"><mml:mi>u<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>v<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mi>R<\/mml:mi><\/mml:math>and fixed integer<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M23\"><mml:mi>n<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:math>.<\/jats:p>","DOI":"10.1155\/2014\/216039","type":"journal-article","created":{"date-parts":[[2014,6,16]],"date-time":"2014-06-16T17:30:28Z","timestamp":1402939828000},"page":"1-8","source":"Crossref","is-referenced-by-count":3,"title":["Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings"],"prefix":"10.1155","volume":"2014","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3213-1254","authenticated-orcid":true,"given":"Vincenzo","family":"De Filippis","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Messina, Viale Ferdinando Stagno d\u2019Alcontres 31, 98166 Messina, Italy"}]},{"given":"Nadeem","family":"UR Rehman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6139-7521","authenticated-orcid":true,"given":"Abu Zaid","family":"Ansari","sequence":"additional","affiliation":[{"name":"Faculty of Science, Islamic University in Madinah, P.O. Box 170, Madinah 42351, Saudi Arabia"}]}],"member":"311","reference":[{"key":"15","doi-asserted-by":"publisher","DOI":"10.1080\/00927879908826682"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/BF02844355"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1155\/S0161171292000255"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1155\/S0161171297000562"},{"issue":"4","key":"1","first-page":"763","volume":"41","year":"2008","journal-title":"Demonstratio Mathematica"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.4134\/CKMS.2012.27.1.069"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1007\/s10474-010-9092-z"},{"issue":"2","key":"9","first-page":"177","volume":"24","year":"1996","journal-title":"Chinese Journal of Mathematics"},{"issue":"1","key":"16","first-page":"27","volume":"20","year":"1992","journal-title":"Bulletin of the Institute of Mathematics. Academia Sinica"},{"key":"11","first-page":"xi+132","year":"1969"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.3906\/mat-0807-27"},{"issue":"1","key":"14","first-page":"13","volume":"5","year":"1998","journal-title":"Algebra Colloquium"},{"issue":"4","key":"19","first-page":"369","volume":"3","year":"1996","journal-title":"Algebra Colloquium"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.2307\/2046841"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(69)90029-5"},{"key":"13","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/BF01670115","volume":"17","year":"1978","journal-title":"Algebra and Logic"},{"key":"4","series-title":"Monographs and Textbooks in Pure and Applied Mathematics","first-page":"xiv+522","volume":"196","year":"1996"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1994.1140"},{"issue":"3","key":"18","first-page":"205","volume":"17","year":"1989","journal-title":"Bulletin of the Institute of Mathematics. Academia Sinica"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2014\/216039.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2014\/216039.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2014\/216039.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T10:57:19Z","timestamp":1498129039000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2014\/216039\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":19,"alternative-id":["216039","216039"],"URL":"https:\/\/doi.org\/10.1155\/2014\/216039","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}