{"title":"素环上的湮灭子条件和广义导数","authors":"B. Dhara, S. Kar, S. Ghosh","doi":"10.1007/s11565-025-00597-x","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>R</i> be a prime ring with char <span>\\((R)\\ne 2\\)</span>, <i>I</i> be a nonzero ideal of <i>R</i>, <span>\\(f(x_1,\\ldots ,x_n)\\)</span> be a noncentral multilinear polynomial over extended centroid <i>C</i> and <span>\\(0\\ne b'\\in R\\)</span>. Denote <span>\\(f(I)=\\{f(t_1,\\ldots ,t_n) | t_1,\\ldots ,t_n\\in I\\}\\)</span>. Suppose that <i>F</i>, <i>G</i> and <i>H</i> are three generalized derivations on <i>R</i> such that </p><div><div><span>$$\\begin{aligned} b'\\Big \\{F\\Big (G(V)V\\Big )-H(V^2)\\Big \\}=0 \\end{aligned}$$</span></div></div><p>for all <span>\\(V\\in f(I)\\)</span>. Then the structure of the maps <i>F</i>, <i>G</i>, <i>H</i> are described. This result naturally generalizes the results obtained by Carini and De Filippis in [Siberian Math. J. 53 (6) (2012), 1051-1060], Dhara and Argac in [Commun. Math. Stat. 4 (2016), 39-54] and completes the incomplete result of Tiwari in [Rend. Circ. Mat. Palermo, II. Ser 71 (2022), 207-223]. At the end of this paper an example is given to show that the primeness condition is not superfluous.</p></div>","PeriodicalId":35009,"journal":{"name":"Annali dell''Universita di Ferrara","volume":"71 2","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2025-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Annihilator conditions and generalized derivations in prime rings\",\"authors\":\"B. Dhara, S. Kar, S. Ghosh\",\"doi\":\"10.1007/s11565-025-00597-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>R</i> be a prime ring with char <span>\\\\((R)\\\\ne 2\\\\)</span>, <i>I</i> be a nonzero ideal of <i>R</i>, <span>\\\\(f(x_1,\\\\ldots ,x_n)\\\\)</span> be a noncentral multilinear polynomial over extended centroid <i>C</i> and <span>\\\\(0\\\\ne b'\\\\in R\\\\)</span>. Denote <span>\\\\(f(I)=\\\\{f(t_1,\\\\ldots ,t_n) | t_1,\\\\ldots ,t_n\\\\in I\\\\}\\\\)</span>. Suppose that <i>F</i>, <i>G</i> and <i>H</i> are three generalized derivations on <i>R</i> such that </p><div><div><span>$$\\\\begin{aligned} b'\\\\Big \\\\{F\\\\Big (G(V)V\\\\Big )-H(V^2)\\\\Big \\\\}=0 \\\\end{aligned}$$</span></div></div><p>for all <span>\\\\(V\\\\in f(I)\\\\)</span>. Then the structure of the maps <i>F</i>, <i>G</i>, <i>H</i> are described. This result naturally generalizes the results obtained by Carini and De Filippis in [Siberian Math. J. 53 (6) (2012), 1051-1060], Dhara and Argac in [Commun. Math. Stat. 4 (2016), 39-54] and completes the incomplete result of Tiwari in [Rend. Circ. Mat. Palermo, II. Ser 71 (2022), 207-223]. At the end of this paper an example is given to show that the primeness condition is not superfluous.</p></div>\",\"PeriodicalId\":35009,\"journal\":{\"name\":\"Annali dell''Universita di Ferrara\",\"volume\":\"71 2\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2025-05-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annali dell''Universita di Ferrara\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11565-025-00597-x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali dell''Universita di Ferrara","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s11565-025-00597-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}
Annihilator conditions and generalized derivations in prime rings
Let R be a prime ring with char \((R)\ne 2\), I be a nonzero ideal of R, \(f(x_1,\ldots ,x_n)\) be a noncentral multilinear polynomial over extended centroid C and \(0\ne b'\in R\). Denote \(f(I)=\{f(t_1,\ldots ,t_n) | t_1,\ldots ,t_n\in I\}\). Suppose that F, G and H are three generalized derivations on R such that
for all \(V\in f(I)\). Then the structure of the maps F, G, H are described. This result naturally generalizes the results obtained by Carini and De Filippis in [Siberian Math. J. 53 (6) (2012), 1051-1060], Dhara and Argac in [Commun. Math. Stat. 4 (2016), 39-54] and completes the incomplete result of Tiwari in [Rend. Circ. Mat. Palermo, II. Ser 71 (2022), 207-223]. At the end of this paper an example is given to show that the primeness condition is not superfluous.
期刊介绍:
Annali dell''Università di Ferrara is a general mathematical journal publishing high quality papers in all aspects of pure and applied mathematics. After a quick preliminary examination, potentially acceptable contributions will be judged by appropriate international referees. Original research papers are preferred, but well-written surveys on important subjects are also welcome.