{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/gmj-2023-2108\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2023-2108","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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}.
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