Generalized derivations of order $2$ on multilinear polynomials in prime rings
Let $R$ be a prime ring of characteristic different from $2$ with a right Martindale quotient ring $Q_r$ and an extended centroid $C$. Let $F$ be a non zero generalized derivation of $R$ and $S$ be the set of evaluations of a non-central valued multilinear polynomial $f(x_1,\ldots,x_n)$ over $C$. Le...
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Ivan Franko National University of Lviv
2022-10-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/38 |
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| author | B. Prajapati C. Gupta |
| author_facet | B. Prajapati C. Gupta |
| author_sort | B. Prajapati |
| collection | DOAJ |
| description | Let $R$ be a prime ring of characteristic different from $2$ with a right Martindale quotient ring $Q_r$ and an extended centroid $C$. Let $F$ be a non zero generalized derivation of $R$ and $S$ be the set of evaluations of a non-central valued multilinear polynomial $f(x_1,\ldots,x_n)$ over $C$. Let $p,q\in R$ be such that
$pF^2(u)u+F^2(u)uq=0$ for all $u\in S$.
Then for all $x\in R$ one of the followings holds:
1) there exists $a\in Q_r$ such that $F(x)=ax$ or $F(x)=xa$ and $a^2=0$,
2) $p=-q\in C$,
3) $f(x_1,\ldots,x_n)^2$ is central valued on $R$ and there exists $a\in Q_r$ such that $F(x)=ax$ with $pa^2+a^2q=0$. |
| format | Article |
| id | doaj-art-955b2b8548184e7caf5e4e515ffe7f52 |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2022-10-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-955b2b8548184e7caf5e4e515ffe7f522025-08-20T03:28:41ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202022-10-01581263510.30970/ms.58.1.26-3538Generalized derivations of order $2$ on multilinear polynomials in prime ringsB. Prajapati0C. Gupta1School of Liberal Studies Dr. B. R. Ambedkar University Delhi, INDIASchool of Liberal Studies Dr. B. R. Ambedkar University Delhi, INDIALet $R$ be a prime ring of characteristic different from $2$ with a right Martindale quotient ring $Q_r$ and an extended centroid $C$. Let $F$ be a non zero generalized derivation of $R$ and $S$ be the set of evaluations of a non-central valued multilinear polynomial $f(x_1,\ldots,x_n)$ over $C$. Let $p,q\in R$ be such that $pF^2(u)u+F^2(u)uq=0$ for all $u\in S$. Then for all $x\in R$ one of the followings holds: 1) there exists $a\in Q_r$ such that $F(x)=ax$ or $F(x)=xa$ and $a^2=0$, 2) $p=-q\in C$, 3) $f(x_1,\ldots,x_n)^2$ is central valued on $R$ and there exists $a\in Q_r$ such that $F(x)=ax$ with $pa^2+a^2q=0$.http://matstud.org.ua/ojs/index.php/matstud/article/view/38prime ringgeneralized derivationextended centroidmartindale's quotient ring |
| spellingShingle | B. Prajapati C. Gupta Generalized derivations of order $2$ on multilinear polynomials in prime rings Математичні Студії prime ring generalized derivation extended centroid martindale's quotient ring |
| title | Generalized derivations of order $2$ on multilinear polynomials in prime rings |
| title_full | Generalized derivations of order $2$ on multilinear polynomials in prime rings |
| title_fullStr | Generalized derivations of order $2$ on multilinear polynomials in prime rings |
| title_full_unstemmed | Generalized derivations of order $2$ on multilinear polynomials in prime rings |
| title_short | Generalized derivations of order $2$ on multilinear polynomials in prime rings |
| title_sort | generalized derivations of order 2 on multilinear polynomials in prime rings |
| topic | prime ring generalized derivation extended centroid martindale's quotient ring |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/38 |
| work_keys_str_mv | AT bprajapati generalizedderivationsoforder2onmultilinearpolynomialsinprimerings AT cgupta generalizedderivationsoforder2onmultilinearpolynomialsinprimerings |