On Lie Ideals and Automorphisms in Prime Rings

N. Rehman

Matematičeskie zametki, Tome 107 (2020) no. 1, pp. 106-111.

Résumé

Let $R$ be a prime ring of characteristic different from $2$ with center $Z$ and extended centroid $C$, and let $L$ be a Lie ideal of $R$. Consider two nontrivial automorphisms $\alpha$ and $\beta$ of $R$ for which there exist integers $m,n\ge 1$ such that $\alpha(u)^n+\beta(u)^m=0$ for all $u\in L$. It is shown that, under these assumptions, either $L$ is central or $R\subseteq M_2(C)$ (where $M_2(C)$ is the ring of $2 \times 2$ matrices over $C$), $L$ is commutative, and $u^{2} \in Z$ for all $u \in L$. In particular, if $L = [R,R]$, then $R$ is commutative.

Détail

Keywords: prime ring, Lie ideal
Mots-clés : automorphism.

Bibliographie

[1] W. S. Martindale III, “Prime rings satisfying a generalized polynomial identity”, J. Algebra, 12 (1969), 576–584 | DOI | MR | Zbl

[2] N. J. Divinsky, “On commuting automorphisms of rings”, Trans. Roy. Soc. Canada Sect. III, 49 (1955), 19–22 | MR | Zbl

[3] E. C. Posner, “Derivations in prime rings”, Proc. Amer. Math. Soc., 8 (1957), 1093–1100 | DOI | MR

[4] J. Luh, “A note on commuting automorphisms of rings”, Amer. Math. Monthly, 77 (1970), 61–62 | DOI | MR | Zbl

[5] J. H. Mayne, “Centralizing automorphisms of prime rings”, Canad. Math. Bull., 19:1 (1976), 113–115 | MR | Zbl

[6] H. E. Bell, W. S. Martindale III, “Centralizing mappings of semiprime rings”, Canad. Math. Bull., 30:1 (1987), 92–101 | DOI | MR | Zbl

[7] C. Lanski, “Left ideals and derivations in semiprime ring”, J. Algebra, 277:2 (2004), 658–667 | DOI | MR | Zbl

[8] A. Giambrun, I. N. Herstein, “Derivations with nilpotent values”, Rend. Circ. Mat. Palermo (2), 30:2 (1981), 199–206 | DOI | MR | Zbl

[9] B. Felzenszwalb, C. Lanski, “On the centralizers of ideals and nil derivations”, J. Algebra, 83:2 (1983), 520–530 | DOI | MR | Zbl

[10] L. Carini, A. Giambruno, “Lie ideals and nil derivations”, Boll. Un. Mat. Ital. A (6), 4:3 (1985), 497–503 | MR | Zbl

[11] C. Lanski, “Derivations with nilpotent values on Lie ideals”, Proc. Am. Math. Soc., 108:1 (1990), 31–37 | DOI | MR | Zbl

[12] L. Carini, V. De Filippis, “Commutators with power central values on a Lie ideal”, Pacfic J. Math., 193:2 (2000), 269–278 | DOI | MR | Zbl

[13] Y. Wang, “Power-centralizing automorphisms of Lie ideals in prime rings”, Comm. Algebra, 34:2 (2006), 609–615 | DOI | MR | Zbl

[14] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with Generalized Identities, Monogr. Textbooks Pure Appl. Math., 196, Marcel Dekker, New York, 1996 | MR | Zbl

[15] C.-L. Chuang, “GPIs having coefficients in Utumi quotient rings”, Proc. Amer. Math. Soc., 103:3 (1988), 723–728 | DOI | MR | Zbl

[16] I. N. Herstein, “Center-like elements in prime rings”, J. Algebra, 60:2 (1979), 567–574 | DOI | MR | Zbl

[17] C. L. Chuang, “Differential identities with automorphisms and antiautomorphisms. II”, J. Algebra, 160:1 (1993), 130–171 | DOI | MR | Zbl

[18] J. Bergen, I. N. Herstein, J. W. Kerr, “Lie ideals and derivations of prime rings”, J. Algebra, 71:1 (1981), 259–267 | MR | Zbl

[19] C. Lanski, S. Montgomery, “Lie structure of prime ring of characteristic 2”, Pacific J. Math., 42:1 (1972), 117–136 | DOI | MR | Zbl

[20] O. M. Di Vincenzo, “On the $n$th centralizer of a Lie ideal”, Boll. Un. Mat. Ital. A (7), 3:1 (1989), 77–85 | MR | Zbl

[21] V. De Filliippis, N. UR Rehman, A. Z. Ansari, “Generalized derivations on power values of Lie ideals in prime and semiprime rings”, Int. J. Math. Math. Sci., 2014, Art. ID 216039 | DOI | MR

[22] N. Jacobson, PI-algebras. An Introduction, Lecture Notes in Math., 441, Springer-Verlag, Berlin, 1975 | DOI | MR | Zbl