An identity on generalized derivations involving multilinear polynomials in prime rings

Proceedings of the Indian Academy of Sciences. Mathematical sciences

C GARG^2¹  B DHARA^1² 

[1] Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India^2;Department ofMathematics, Belda College, Belda, Paschim Medinipur 721 424, India^1

关键词: Derivation;    generalized derivation;    prime ring;    extended centroid;    Utumi quotient ring;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let $R$ be a prime ring of characteristic different from $2$ with its Utumi ring of quotients $U$, extended centroid $C$, $f(x_{1},\ldots,x_{n})$ a multilinear polynomial over $C$, which is not central-valued on $R$ and $d$ a nonzero derivation of $R$. By $f(R)$, we mean the set of all evaluations of the polynomial $f(x_{1},\ldots,x_{n})$ in $R$. In the present paper, we study $b[d(u),u]+p[d(u),u]q+[d(u),u]c=0$ for all $u\in f(R)$, which includes left-sided, right-sided as well as two-sided annihilating conditions of the set $\{[d(u),u] : u\in f(R)\}$.We also examine some consequences of this result related to generalized derivations and we prove that if $F$ is a generalized derivation of $R$ and $d$ is a nonzero derivation of $R$ such that $$F^{2}([d(u), u])=0$$ for all $u\in f(R)$, then there exists $a\in U$ with $a^{2}=0$ such that $F(x)=xa$ for all $x\in R$ or $F(x)=ax$ for all $x\in R$.

【 授权许可】

CC BY