{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:38:59Z","timestamp":1753886339190,"version":"3.41.2"},"reference-count":12,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,2,25]],"date-time":"2012-02-25T00:00:00Z","timestamp":1330128000000},"content-version":"vor","delay-in-days":55,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["10871056","HEUCF20111132"],"award-info":[{"award-number":["10871056","HEUCF20111132"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["10871056","HEUCF20111132"],"award-info":[{"award-number":["10871056","HEUCF20111132"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Let <jats:italic>A<\/jats:italic> be an algebra over a commutative unital ring <jats:italic>\ud835\udc9e<\/jats:italic>. We say that <jats:italic>A<\/jats:italic> is zero triple\nproduct determined if for every <jats:italic>\ud835\udc9e<\/jats:italic>\u2010module <jats:italic>X<\/jats:italic> and every trilinear map {\u00b7, \u00b7, \u00b7}, the following\nholds: if {<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>, <jats:italic>z<\/jats:italic>} = 0 whenever <jats:italic>x<\/jats:italic><jats:italic>y<\/jats:italic><jats:italic>z<\/jats:italic> = 0, then there exists a <jats:italic>\ud835\udc9e<\/jats:italic>\u2010linear operator <jats:italic>T<\/jats:italic> : <jats:italic>A<\/jats:italic><jats:sup>3<\/jats:sup> \u2192 <jats:italic>X<\/jats:italic>\nsuch that {<jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>, <jats:italic>z<\/jats:italic>} = <jats:italic>T<\/jats:italic>(<jats:italic>x<\/jats:italic><jats:italic>y<\/jats:italic><jats:italic>z<\/jats:italic>) for all <jats:italic>x<\/jats:italic>, <jats:italic>y<\/jats:italic>, <jats:italic>z<\/jats:italic> \u2208 <jats:italic>A<\/jats:italic>. If the ordinary\ntriple product in the aforementioned definition is replaced by Jordan triple\nproduct, then <jats:italic>A<\/jats:italic> is called zero Jordan triple\nproduct determined. This paper mainly shows that matrix algebra <jats:italic>M<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>B<\/jats:italic>), <jats:italic>n<\/jats:italic> \u2265 3, where\n<jats:italic>B<\/jats:italic> is any commutative unital algebra even different\nfrom the above mentioned commutative unital algebra <jats:italic>\ud835\udc9e<\/jats:italic>, is always zero triple product determined, and <jats:italic>M<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>F<\/jats:italic>), <jats:italic>n<\/jats:italic> \u2265 3, where <jats:italic>F<\/jats:italic>\nis any field with ch<jats:italic>F<\/jats:italic> \u2260 2, is also zero Jordan triple product determined.<\/jats:p>","DOI":"10.1155\/2012\/925092","type":"journal-article","created":{"date-parts":[[2012,2,25]],"date-time":"2012-02-25T22:23:07Z","timestamp":1330208587000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Zero Triple Product Determined Matrix Algebras"],"prefix":"10.1155","volume":"2012","author":[{"given":"Hongmei","family":"Yao","sequence":"first","affiliation":[]},{"given":"Baodong","family":"Zheng","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,2,25]]},"reference":[{"doi-asserted-by":"publisher","key":"e_1_2_5_1_2","DOI":"10.1016\/j.laa.2007.11.018"},{"doi-asserted-by":"publisher","key":"e_1_2_5_2_2","DOI":"10.1017\/S0308210505000090"},{"doi-asserted-by":"publisher","key":"e_1_2_5_3_2","DOI":"10.4064\/sm193-2-3"},{"doi-asserted-by":"publisher","key":"e_1_2_5_4_2","DOI":"10.1017\/S0013091509000534"},{"doi-asserted-by":"publisher","key":"e_1_2_5_5_2","DOI":"10.1016\/j.jalgebra.2005.11.002"},{"doi-asserted-by":"publisher","key":"e_1_2_5_6_2","DOI":"10.1080\/03081080903191672"},{"doi-asserted-by":"publisher","key":"e_1_2_5_7_2","DOI":"10.1016\/j.laa.2006.11.013"},{"key":"e_1_2_5_8_2","first-page":"1","article-title":"Maps preserving the idempotency of products of operators.","author":"Petek T.","year":"2010","journal-title":"Linear and Multilinear Algebra"},{"doi-asserted-by":"publisher","key":"e_1_2_5_9_2","DOI":"10.1016\/j.laa.2007.04.017"},{"doi-asserted-by":"publisher","key":"e_1_2_5_10_2","DOI":"10.1016\/j.laa.2003.06.004"},{"doi-asserted-by":"publisher","key":"e_1_2_5_11_2","DOI":"10.7146\/math.scand.a-12160"},{"doi-asserted-by":"publisher","key":"e_1_2_5_12_2","DOI":"10.1016\/j.laa.2011.04.033"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/925092.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/925092.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2012\/925092","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T06:52:20Z","timestamp":1718088740000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2012\/925092"}},"subtitle":[],"editor":[{"given":"Xianhua","family":"Tang","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1155\/2012\/925092"],"URL":"https:\/\/doi.org\/10.1155\/2012\/925092","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2012,1]]},"assertion":[{"value":"2011-08-09","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-12-20","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-02-25","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"925092"}}