On the trace of permuting tri-derivations on rings

  • D. Yılmaz Erzurum Technical University, Faculty of Science, Department of Mathematics Erzurum, Turkey
  • H. Yazarlı Sivas Cumhuriyet University, Faculty of Science, Department of Mathematics Sivas, Turkey
Keywords: derivation;, trace of permuting tri-derivation;, endomorphism

Abstract

In the paper we examined the some effects of derivation, trace of permuting tri-derivation and endomorphism on each other in prime and semiprime ring.
Let $R$ be a $2,3$-torsion free prime ring and $F:R\times R\times R\rightarrow R$ be a permuting tri-derivation with trace $f$, $ d:R\rightarrow R$ be a derivation. In particular, the following assertions have been proved:
1) if $[d(r),r]=f(r)$ for all $r\in R$, then $R$ is commutative or $d=0$ (Theorem 1);\

2) if $g:R\rightarrow R$ is an endomorphism such that $F(d(r),r,r)=g(r)$ for all $r\in R$, then $F=0$ or $d=0$ (Theorem 2);

3) if $F(d(r),r,r)=f(r)$ for all $r\in R$, then $(i)$ $F=0$ or $d=0$, $(ii)$ $d(r)\circ f(r)=0$ for all $r\in R$ (Theorem 3).

In the other hand, if there exist permuting tri-derivations $F_{1},F_{2}:R\times R\times R\rightarrow R$ such that $F_{1}(f_{2}(r),r,r)=f_{1}(r)$ for all $r\in R$, where $f_{1}$ and $%f_{2}$ are traces of $F_{1}$ and $F_{2}$, respectively, then $(i)$ $F_{1}=0$ or $F_{2}=0$, $(ii)$ $f_{1}(r)\circ f_{2}(r)=0$ for all $r\in R$ (Theorem 4).

Author Biographies

D. Yılmaz, Erzurum Technical University, Faculty of Science, Department of Mathematics Erzurum, Turkey

Erzurum Technical University, Faculty of Science, Department of Mathematics
Erzurum, Turkey

H. Yazarlı, Sivas Cumhuriyet University, Faculty of Science, Department of Mathematics Sivas, Turkey

Sivas Cumhuriyet University, Faculty of Science, Department of Mathematics
Sivas, Turkey

References

H. Durna, S. Oguz, Permuting tri-derivations in prime and semi-prime rings, International Journal of Algebra and Statistics, 5 (2016), №1, 52–58.

U. Leerawat and S. Khun-in, On Trace of Symmetric Bi-derivations on Rings, International Journal of Mathematics and Computer Science, 16 (2021), №2, 743–752.

J. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull., 27 (1984) №1, 122–126.

D. Ozden, M. A. Ozturk, Y. B. Jun, Permuting tri-derivations in prime and semi-prime gamma rings, Kyungpook Mathematical Journal, 46 (2006), №2, 153–167.

M.A. Ozturk, Permuting Tri-derivations in prime and semiprime rings, East Asian Math. J., 15 (1999), №2, 177–190.

E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), №6, 1093–1100.

H. Yazarli, M.A. Ozturk, Y.B. Jun, Tri-additive maps and permuting tri-derivations, Commun. Fac. Sci. Univ. Ankara. Ser. A1, Mathematics and Statistics, 54 (2005), №01, 1–14.

H. Yazarli, Permuting triderivations of prime and semiprime rings, Miskolc Mathematical Notes, 18 (2017), №1, 489–497.

Published
2022-10-31
How to Cite
Yılmaz, D., & Yazarlı, H. (2022). On the trace of permuting tri-derivations on rings. Matematychni Studii, 58(1), 20-25. https://doi.org/10.30970/ms.58.1.20-25
Section
Articles