Two generalized derivations on Lie ideals in prime rings

Yıl 2023, Cilt: 34 Sayı: 34, 48 - 61, 10.07.2023

Ashutosh Pandey Balchand Prajapati

https://doi.org/10.24330/ieja.1281636

Öz

Let $R$ be a prime ring of characteristic not equal to $2$, $U$ be the Utumi quotient ring of $R$ and $C$ be the extended centroid of $R$. Let $G$ and $F$ be two generalized derivations on $R$ and $L$ be a non-central Lie ideal of $R$. If $F\Big(G(u)\Big)u = G(u^{2})$ for all $u \in L$, then one of the following holds:

(1) $G=0$.
(2) There exist $p,q \in U$ such that $G(x)=p x$, $F(x)=qx$ for all $x \in R$ with $qp=p$.
(3) $R$ satisfies $s_4$.

Anahtar Kelimeler

Lie ideal, generalized derivation, extended centroid, Utumi quotient ring

Kaynakça

• N. Argaç, L. Carini and V. De Filippis, An Engel condition with generalized derivations on Lie ideals, Taiwanese J. Math., 12(2) (2008), 419-433.
• K. I. Beidar, Rings with generalized identities III, Vestnik Moskov. Univ. Ser. I Mat. Mekh, 4 (1978), 66-73.
• K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with Generalized Identities, CRC Press, 1995.
• M. Bresar, Centralizing mappings and derivations in prime rings, J. Algebra, 156(2) (1993), 385-394. C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103(3) (1988), 723-728. V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra, 40(6) (2012), 1918-1932.
• B. Dhara, Annihilator condition on power values of derivations, Indian J. Pure Appl. Math., 42 (2011), 357-369. O. M. Di Vincenzo, On the n-th centralizer of a Lie ideal, Boll. Un. Mat. Ital. A (7), 3 (1989), 77-85. T. Erickson, W. S. Martindale and J. Osborn, Prime nonassociative algebras, Pacific J. Math., 60(1) (1975), 49-63.
• C. Faith and Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hungar., 14 (1963), 369-371.
• I. N. Herstein, Topics in Ring Theory, University of Chicago Press, Chicago, 1969.
• N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications, 1956. V. K. Kharchenko, Differential identities of prime rings, Algebra i Logika, 17(2) (1978), 155-168. C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc., 118(3) (1993), 731-734. C. Lanski and S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math., 42 (1972), 117-136.
• T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20(1) (1992), 27-38.
• T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra, 27(8) (1999), 4057-4073.
• W. S. Martindale, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584.
• E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.
• E. C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc., 11 (1960), 180-183. S. K. Tiwari, Identities with generalized derivations in prime rings, Rend. Circ. Mat. Palermo (2), 71 (2022), 207-223.