Pair of generalized derivations acting on multilinear polynomials in prime rings

Year 2020, Volume: 49 Issue: 2, 740 - 753, 02.04.2020

Basudeb Dhara , Sukhendu Kar , Priyadwip Das

https://doi.org/10.15672/hujms.588747

Cited By: 2

Abstract

Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$ and $f(r_1,\ldots,r_n)$ be a multilinear polynomial over $C$, which is not central valued on $R$. Suppose that $F$ and $G$ are two nonzero generalized derivations of $R$ such that $G\neq Id$ (identity map) and $$F(f(r)^2)=F(f(r))G(f(r))+G(f(r))F(f(r))$$

for all $r=(r_1,\ldots,r_n)\in R^n$. Then one of the following holds:

(1) there exist $\lambda \in C$ and $\mu \in C$ such that $F(x)=\lambda x$ and $G(x)=\mu x$ for
all $x\in R$ with $2\mu=1$;
(2) there exist $\lambda \in C$ and $p,q\in U$ such that $F(x)=\lambda x$ and $G(x)=px+xq$ for all $x\in R$ with $p+q\in C$,
$2(p+q)=1$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;
(3) there exist $\lambda \in C$ and $a\in U$ such that $F(x)=[a,x]$ and $G(x)=\lambda x$ for all $x\in R$ with $f(x_1,\ldots,x_n)^2$ is central valued on $R$;
(4) there exist $\lambda \in C$ and $a,b\in U$ such that $F(x)=ax+xb$ and $G(x)=\lambda x$ for all $x\in R$ with $a+b\in C$, $2\lambda =1$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;
(5) there exist $a, p\in U$ such that $F(x)=xa$ and $G(x)=px$ for all $x\in R$, with $(p-1)a=-ap\in C$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$;
(6) there exist $a, q\in U$ such that $F(x)=ax$ and $G(x)=xq$ for all $x\in R$ with $a(q-1)=-qa\in C$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$.

Keywords

prime ring, derivation, generalized derivation, extended centroid

References

• [1] N. Argac and V. De Filippis, Actions of generalized derivations on multilinear polynomials in prime rings, Algebra Colloq. 18 (Spec 01), 955–964, 2011.
• [2] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra, 156, 385–394, 1993.
• [3] C.L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (3), 723–728, 1988.
• [4] V. De Filippis, O.M. Di Vincenzo, and C.Y. Pan, Quadratic central differential identities on a multilinear polynomial, Comm. Algebra, 36 (10), 3671–3681, 2008.
• [5] V. De Filippis and O.M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra, 40, 1918–1932, 2012.
• [6] B. Dhara, S. Kar, and K.G. Pradhan, Identities with generalized derivations on multilinear polynomials in prime rings, Afr. Mat. 27, 1347–1360, 2016.
• [7] M. Fosner and J. Vukman, Identities with generalized derivations in prime rings, Mediter. J. Math. 9 (4), 847–863, 2012.
• [8] T.S. Erickson, W.S. Martindale III, and J.M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60, 49–63, 1975.
• [9] C. Faith and Y. Utumi, On a new proof of Litoff’s theorem, Acta Math. Acad. Sci. Hung. 14, 369–371, 1963.
• [10] N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Pub. 37, Amer. Math. Soc., Providence, RI, 1964.
• [11] V.K. Kharchenko, Differential identity of prime rings, Algebra Logic, 17, 155–168, 1978.
• [12] T.K. Lee, Generalized derivations of left faithful rings, Comm. Algebra, 27 (8), 4057– 4073, 1999.
• [13] T.K. Lee and W.K. Shiue, Derivations co-centralizing polynomials, Taiwanese J. Math. 2 (4), 457–467, 1998.
• [14] T.K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20 (1), 27–38, 1992.
• [15] U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202, 297–103, 1975.
• [16] W.S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12, 576–584, 1969.
• [17] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093–1100, 1957.
• [18] F. Rania and G. Scudo, A quadratic differential identity with generalized derivations on multilinear polynomials in prime rings, Mediterr. J. Math. 11, 273–285, 2014.
• [19] N.B. Yarbil and V. De Filippis, A quadratic differential identity with skew derivations, Comm. Algebra, 46 (1), 205–216, 2018.