{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T14:50:51Z","timestamp":1746802251452,"version":"3.40.5"},"reference-count":11,"publisher":"Wiley","license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2016]]},"abstract":"<jats:p>We introduce the notion of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-derivation as a new derivation of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>-algebra. For an endomorphism map<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><\/mml:math>of any<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>-algebra<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>X<\/mml:mi><mml:mo>,<\/mml:mo><\/mml:math>we show that at least one<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-derivation of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>exists. Moreover, for such a map, we show that a self-map<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>d<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>is<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\"><mml:mrow><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-derivation of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>if<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>is an associative medial<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>-algebra. For a medial<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>-algebra<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>,<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>d<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>is<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\"><mml:mrow><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-derivation of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M20\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>if<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M21\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>d<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:math>is an outside<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M22\"><mml:mrow><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-derivation of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M23\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>. Finally, we show that if<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M24\"><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><\/mml:math>is the identity endomorphism of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M25\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>then the composition of two<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M26\"><mml:mrow><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-derivations of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M27\"><mml:mrow><mml:mi>X<\/mml:mi><\/mml:mrow><\/mml:math>is a<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M28\"><mml:mrow><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-derivation. Moreover, we give a condition to get a commutative composition.<\/jats:p>","DOI":"10.1155\/2016\/9276096","type":"journal-article","created":{"date-parts":[[2016,4,28]],"date-time":"2016-04-28T17:23:55Z","timestamp":1461864235000},"page":"1-5","source":"Crossref","is-referenced-by-count":0,"title":["<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msub><mml:mrow><mml:mi>f<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>-Derivations of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>G<\/mml:mi><\/mml:mrow><\/mml:math>-Algebra"],"prefix":"10.1155","volume":"2016","author":[{"given":"Deena","family":"Al-Kadi","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistic, Faculty of Science, Taif University, P.O. Box 888, Taif 21974, Saudi Arabia"}]}],"member":"311","reference":[{"issue":"1-2","key":"1","first-page":"13","volume":"46","year":"2006","journal-title":"The Journal of Natural Sciences and Mathematics"},{"issue":"2","key":"2","first-page":"398","volume":"43","year":"2009","journal-title":"Kasetsart Journal"},{"doi-asserted-by":"publisher","key":"3","DOI":"10.4197\/sci.22-1.5"},{"doi-asserted-by":"publisher","key":"4","DOI":"10.1016\/j.ins.2003.03.001"},{"issue":"3","key":"5","first-page":"435","volume":"33","year":"2007","journal-title":"Soochow Journal of Mathematics"},{"doi-asserted-by":"publisher","key":"6","DOI":"10.1155\/ijmms.2005.1675"},{"doi-asserted-by":"publisher","key":"7","DOI":"10.4134\/ckms.2009.24.3.321"},{"doi-asserted-by":"publisher","key":"8","DOI":"10.1155\/2012\/403209"},{"doi-asserted-by":"publisher","key":"9","DOI":"10.1155\/2012\/872784"},{"doi-asserted-by":"publisher","key":"10","DOI":"10.12988\/ams.2013.24298"},{"issue":"3","key":"11","first-page":"1","volume":"8","year":"2012","journal-title":"Scientia Magna"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2016\/9276096.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2016\/9276096.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2016\/9276096.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,26]],"date-time":"2016-07-26T13:28:58Z","timestamp":1469539738000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2016\/9276096\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"references-count":11,"alternative-id":["9276096","9276096"],"URL":"https:\/\/doi.org\/10.1155\/2016\/9276096","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2016]]}}}