### Some Identities in Rings and Near-Rings with Derivations

Download PDF

**Authors: ** A. BOUA

**DOI: ** 10.46793/KgJMat2101.075B

**Abstract: **

In the present paper we investigate commutativity in prime rings and 3-prime near-rings admitting a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings and 3-prime near-rings have been generalized.

**Keywords: **

3-prime near-ring, prime ring, derivations, commutativity, left multiplier.

**References: **

[1] M. Ashraf, A. Ali and S. Ali, (σ,τ)-derivations on prime near-rings, Arch. Math. (Brno) 40 (2004), 281–286.

[2] M. Ashraf and S. Ali, On (σ,τ)-derivations of prime near-rings II, Sarajevo J. Math. 4 (2008), 23–30.

[3] D. Basudeb, Remarks on generalized derivations in prime and semiprime rings, Int. J. Math. Math. Sci. (2010), Article ID 646587.

[4] K. I. Beidar, Y. Fong and X. K. Wang, Posner and Herstein theorems for derivations of 3-prime near-rings, Comm. Algebra 24 (1996), 1581–1589.

[5] H. E. Bell, On Derivations in Near-Rings II, Kluwer Academic Publishers, Amsterdam, 1997.

[6] H. E. Bell and G. Mason, On Derivations in Near-Ring in Near-Rings and Near-Fields, North-Holland, Amsterdam, 1987, 31–35.

[7] H. E . Bell and G. Mason, On derivations in near-rings and rings, Math. J. Okayama Univ. 34 (1992), 135–144.

[8] H. E. Bell, A. Boua and L. Oukhtite, On derivations of prime near-rings, Afr. Diaspora J. Math. 14(1) (2012), 65–72.

[9] A. Boua, A. Y. Abdelwanis and A. Chillali, Some commutativity theorems for near-rings with left multipliers, Kragujevac J. Math. 44(2) (2020), 205–216.

[10] M. Brešar, On the distance of composition of two derivations to the generalized derivations, Glasg. Math. J. 33 (1991), 89–93.

[11] M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci. 15 (1992), 205–206.

[12] Ö. Gölbasi and E. Koc, Notes on commutativity of prime rings with generalized derivation, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 58 (2009), 39–46.

[13] Ö. Gölbasi and E. Koc, Some commutativity theorems of prime rings with generalized (σ,τ)-derivation, Commun. Korean Math. Soc. 26 (2011), 445–454.

[14] Ö. Gölbasi, Notes on prime near-rings with generalized derivation, Southeast Asian Bull. Math. 30 (2006), 49–54.

[15] A. A. M. Kamal, σ-Derivations on prime near-rings, Tamkang J. Math. 32 (2001), 89–93.

[16] M. A. Quadri, M. Shadab Khan and N. Rehman, Generalized derivation and commutativity of prime rings, Indian J. Pure Appl. Math. 34(9) (2003), 1393–1396.

[17] R. Raina, V. K. Bhat and N. Kumari, Commutativity of prime Γ-near-rings with (σ,τ)-derivation, Acta Math. Acad. Paedagog. Nyhazi 25 (2009), 165–173.

[18] N. Ur-Rehman, On commutativity of rings with generalized derivations, Math. J. Okayama Univ. 44 (2002), 43–49.

[19] Y. Shang, A study of derivations in prime near-rings 1, Math. Balkanica (N.S.) 25(4) (2011), 413–418.

[20] X. K. Wang, Derivations in prime near-rings, Proc. Amer. Math. Soc. 121 (1994), 361–366.