{"title":"沿理想的混杂代数上的广义推导","authors":"Brahim Boudine, Mohammed Zerra","doi":"10.1515/gmj-2023-2108","DOIUrl":null,"url":null,"abstract":"Let <jats:italic>A</jats:italic> and <jats:italic>B</jats:italic> be two associative rings, let <jats:italic>I</jats:italic> be an ideal of <jats:italic>B</jats:italic> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>Hom</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0167.png\" /> <jats:tex-math>{f\\in\\mathrm{Hom}(A,B)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we give a complete description of generalized derivations over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0101.png\" /> <jats:tex-math>{A\\bowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Furthermore, when <jats:italic>A</jats:italic> is prime or semi-prime, we give several identities on generalized derivations which provide the commutativity of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0101.png\" /> <jats:tex-math>{A\\bowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"6 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generalized derivations over amalgamated algebras along an ideal\",\"authors\":\"Brahim Boudine, Mohammed Zerra\",\"doi\":\"10.1515/gmj-2023-2108\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <jats:italic>A</jats:italic> and <jats:italic>B</jats:italic> be two associative rings, let <jats:italic>I</jats:italic> be an ideal of <jats:italic>B</jats:italic> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>Hom</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2108_eq_0167.png\\\" /> <jats:tex-math>{f\\\\in\\\\mathrm{Hom}(A,B)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we give a complete description of generalized derivations over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2108_eq_0101.png\\\" /> <jats:tex-math>{A\\\\bowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Furthermore, when <jats:italic>A</jats:italic> is prime or semi-prime, we give several identities on generalized derivations which provide the commutativity of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2108_eq_0101.png\\\" /> <jats:tex-math>{A\\\\bowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":55101,\"journal\":{\"name\":\"Georgian Mathematical Journal\",\"volume\":\"6 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Georgian Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/gmj-2023-2108\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Georgian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2023-2108","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让 A 和 B 是两个关联环，让 I 是 B 的一个理想，让 f ∈ Hom ( A , B ) {f\inmathrm{Hom}(A,B)} 。在本文中，我们将完整地描述 A ⋈ f I {A\bowtie^{f}I} 上的广义推导。此外，当 A 是质数或半质数时，我们给出了关于广义推导的几个同素异形，这些同素异形提供了 A ⋈ f I {A\bowtie^{f}I} 的交换性。
Generalized derivations over amalgamated algebras along an ideal
Let A and B be two associative rings, let I be an ideal of B and let f∈Hom(A,B){f\in\mathrm{Hom}(A,B)}. In this paper, we give a complete description of generalized derivations over A⋈fI{A\bowtie^{f}I}. Furthermore, when A is prime or semi-prime, we give several identities on generalized derivations which provide the commutativity of A⋈fI{A\bowtie^{f}I}.
期刊介绍：
The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter. The concern of this international journal is the publication of research articles of best scientific standard in pure and applied mathematics. Special emphasis is put on the presentation of results obtained by Georgian mathematicians.
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