素环中涉及广义派生的集中化等式

IF 0.8 4区 数学 Q2 MATHEMATICS
Vincenzo De Filippis, Pallavee Gupta, Shailesh Kumar Tiwari, Balchand Prajapati
{"title":"素环中涉及广义派生的集中化等式","authors":"Vincenzo De Filippis, Pallavee Gupta, Shailesh Kumar Tiwari, Balchand Prajapati","doi":"10.1515/gmj-2023-2109","DOIUrl":null,"url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /> <jats:tex-math>{\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a prime ring of characteristic not equal to 2, let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">𝒰</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0527.png\" /> <jats:tex-math>{\\mathcal{U}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be Utumi quotient ring of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /> <jats:tex-math>{\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">𝒞</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0481.png\" /> <jats:tex-math>{\\mathcal{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the extended centroid of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /> <jats:tex-math>{\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let Δ be a generalized derivation on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /> <jats:tex-math>{\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>δ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0428.png\" /> <jats:tex-math>{\\delta_{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>δ</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0440.png\" /> <jats:tex-math>{\\delta_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be derivations on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /> <jats:tex-math>{\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>v</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0622.png\" /> <jats:tex-math>{p(v)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a multilinear polynomial on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /> <jats:tex-math>{\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is non-central valued on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /> <jats:tex-math>{\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. If <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi>δ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>v</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>v</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>δ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>v</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <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-2023-2109_eq_0411.png\" /> <jats:tex-math>{\\delta_{1}(\\Delta^{2}(p(v))p(v))=\\delta_{2}(\\Delta(p(v)^{2}))}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi mathvariant=\"script\">ℛ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0668.png\" /> <jats:tex-math>{v\\in\\mathcal{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then we find the complete description of Δ, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>δ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0428.png\" /> <jats:tex-math>{\\delta_{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>δ</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0440.png\" /> <jats:tex-math>{\\delta_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"33 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Centralizing identities involving generalized derivations in prime rings\",\"authors\":\"Vincenzo De Filippis, Pallavee Gupta, Shailesh Kumar Tiwari, Balchand Prajapati\",\"doi\":\"10.1515/gmj-2023-2109\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0519.png\\\" /> <jats:tex-math>{\\\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a prime ring of characteristic not equal to 2, let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">𝒰</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0527.png\\\" /> <jats:tex-math>{\\\\mathcal{U}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be Utumi quotient ring of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0519.png\\\" /> <jats:tex-math>{\\\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">𝒞</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0481.png\\\" /> <jats:tex-math>{\\\\mathcal{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the extended centroid of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0519.png\\\" /> <jats:tex-math>{\\\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let Δ be a generalized derivation on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0519.png\\\" /> <jats:tex-math>{\\\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>δ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0428.png\\\" /> <jats:tex-math>{\\\\delta_{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>δ</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0440.png\\\" /> <jats:tex-math>{\\\\delta_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be derivations on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0519.png\\\" /> <jats:tex-math>{\\\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>v</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0622.png\\\" /> <jats:tex-math>{p(v)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a multilinear polynomial on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0519.png\\\" /> <jats:tex-math>{\\\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which is non-central valued on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi mathvariant=\\\"script\\\">ℛ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0519.png\\\" /> <jats:tex-math>{\\\\mathcal{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. If <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mrow> <m:msub> <m:mi>δ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:msup> <m:mi mathvariant=\\\"normal\\\">Δ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>v</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>v</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>δ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:mi mathvariant=\\\"normal\\\">Δ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>v</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> <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-2023-2109_eq_0411.png\\\" /> <jats:tex-math>{\\\\delta_{1}(\\\\Delta^{2}(p(v))p(v))=\\\\delta_{2}(\\\\Delta(p(v)^{2}))}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi mathvariant=\\\"script\\\">ℛ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0668.png\\\" /> <jats:tex-math>{v\\\\in\\\\mathcal{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then we find the complete description of Δ, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>δ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0428.png\\\" /> <jats:tex-math>{\\\\delta_{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi>δ</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2023-2109_eq_0440.png\\\" /> <jats:tex-math>{\\\\delta_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":55101,\"journal\":{\"name\":\"Georgian Mathematical Journal\",\"volume\":\"33 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-2023-2109\",\"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-2023-2109","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

设 ℛ {\mathcal{R}} 是一个特征不等于 2 的素环,设 𝒰 {\mathcal{U}} 是 ℛ {\mathcal{R}} 的乌图米商环,设 𝒞 {\mathcal{C}} 是 ℛ {\mathcal{R}} 的扩展中心点。 .设 Δ 是ℛ {\mathcal{R}} 上的广义推导。 让 δ 1 {\delta_{1}} 和 δ 2 {\delta_{2}} 是ℛ {\mathcal{R}} 上的导数。 .设 p ( v ) {p(v)} 是ℛ {mathcal{R}} 上的多线性多项式。 上的非中心值。 .如果 δ 1 ( Δ 2 ( p ( v ) ) p ( v ) ) = δ 2 ( Δ ( p ( v ) 2 ) ) {\delta_{1}(\Delta^{2}(p(v))p(v))=\delta_{2}(\Delta(p(v)^{2}))} 对于所有 v∈ ℛ n {v\in\mathcal{R}^{n}} ,那么我们就能找到完整的描述。 那么我们就能找到对 Δ、δ 1 {delta_{1}} 和 δ 2 {delta_{2}} 的完整描述。 .
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Centralizing identities involving generalized derivations in prime rings
Let {\mathcal{R}} be a prime ring of characteristic not equal to 2, let 𝒰 {\mathcal{U}} be Utumi quotient ring of {\mathcal{R}} and let 𝒞 {\mathcal{C}} be the extended centroid of {\mathcal{R}} . Let Δ be a generalized derivation on {\mathcal{R}} , and let δ 1 {\delta_{1}} and δ 2 {\delta_{2}} be derivations on {\mathcal{R}} . Let p ( v ) {p(v)} be a multilinear polynomial on {\mathcal{R}} , which is non-central valued on {\mathcal{R}} . If δ 1 ( Δ 2 ( p ( v ) ) p ( v ) ) = δ 2 ( Δ ( p ( v ) 2 ) ) {\delta_{1}(\Delta^{2}(p(v))p(v))=\delta_{2}(\Delta(p(v)^{2}))} for all v n {v\in\mathcal{R}^{n}} , then we find the complete description of Δ, δ 1 {\delta_{1}} and δ 2 {\delta_{2}} .
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来源期刊
CiteScore
1.70
自引率
0.00%
发文量
76
审稿时长
>12 weeks
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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