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γ-Lie structures in γ-prime gamma rings with derivations

Year 2015, Volume: 2 Issue: 1, 25 - 37, 22.01.2015
https://doi.org/10.13069/jacodesmath.87481

Abstract

Let $M$ be a $\gamma$-prime weak Nobusawa $\Gamma $-ring and $d\neq 0$ be a $k$-derivation of $M$ such that $k\left( \gamma \right) =0$ and $U$ be a $\gamma$-Lie ideal of $M$. In this paper, we introduce definitions of $\gamma$-subring, $\gamma$-ideal, $\gamma$-prime $\Gamma$-ring and $\gamma$-Lie ideal of M and prove that if $U\nsubseteq C_{\gamma}$, $char$M$\neq2$ and $d^3\neq0$, then the $\gamma$-subring generated by $d(U)$ contains a nonzero ideal of $M$. We also prove that if $[u,d(u)]_{\gamma}\in C_{\gamma}$ for all $u\in U$, then $U$ is contained in the $\gamma$-center of $M$ when char$M\neq2$ or $3$. And if $[u,d(u)]_{\gamma}\in C_{\gamma}$ for all $u\in U$ and $U$ is also a $\gamma$-subring, then $U$ is $\gamma$-commutative when char$M=2$.

References

  • R. Awtar, Lie and Jordan structure in prime rings with derivations, Proc. Amer. Math. Soc., 41, 67-74, 1973.
  • W. E. Barnes, On the Γ-rings of Nobusawa, Pacific J. Math., 18, 411-422, 1966.
  • J. Bergen, J.W. Kerr, I.N. Herstein, Lie ideals and derivations of prime rings, J. Algebra, 71, 259-267, 1981.
  • I. N. Herstein, A note on derivations, Canad. Math. Bull., 21(3), 369-370, 1978.
  • I. N. Herstein, A note on derivations II, Canad. Math. Bull., 22(4), 509-511, 1979.
  • I. N. Herstein, Topics in Ring Theory, The Univ. of Chicago Press, 1969.
  • H. Kandamar, The k-Derivation of a Gamma-Ring, Turk. J. Math., 23(3), 221-229, 2000.
  • A. R. Khan, M. A. Chaudhry, I. Javaid, Generalized Derivations on Prime Γ-Rings, World Appl. Sci. J., 23(12), 59-64, 2013.
  • S. Kyuno, Gamma Rings, Hadronic Press, 1991.
  • N. Nobusawa, On a generalization of the ring theory, Osaka J. Math., 1, 81-89, 1964.
  • E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8, 1093-1100, 1957.
  • N. N. Suliman, A. H. Majeed, Lie Ideals in Prime Γ-Rings with Derivations, Discussiones Mathe
  • maticae, 33, 49-56, 2013.

γ-Lie structures in γ-prime gamma rings with derivations

Year 2015, Volume: 2 Issue: 1, 25 - 37, 22.01.2015
https://doi.org/10.13069/jacodesmath.87481

Abstract

Let $M$ be a $\gamma$-prime weak Nobusawa $\Gamma $-ring and $d\neq 0$ be a $k$-derivation of $M$ such that $k\left( \gamma \right) =0$ and $U$ be a $\gamma$-Lie ideal of $M$. In this paper, we introduce definitions of $\gamma$-subring, $\gamma$-ideal, $\gamma$-prime $\Gamma$-ring and $\gamma$-Lie ideal of M and prove that if $U\nsubseteq C_{\gamma}$, $char$M$\neq2$ and $d^3\neq0$, then the $\gamma$-subring generated by $d(U)$ contains a nonzero ideal of $M$. We also prove that if $[u,d(u)]_{\gamma}\in C_{\gamma}$ for all $u\in U$, then $U$ is contained in the $\gamma$-center of $M$ when char$M\neq2$ or $3$. And if $[u,d(u)]_{\gamma}\in C_{\gamma}$ for all $u\in U$ and $U$ is also a $\gamma$-subring, then $U$ is $\gamma$-commutative when char$M=2$.

References

  • R. Awtar, Lie and Jordan structure in prime rings with derivations, Proc. Amer. Math. Soc., 41, 67-74, 1973.
  • W. E. Barnes, On the Γ-rings of Nobusawa, Pacific J. Math., 18, 411-422, 1966.
  • J. Bergen, J.W. Kerr, I.N. Herstein, Lie ideals and derivations of prime rings, J. Algebra, 71, 259-267, 1981.
  • I. N. Herstein, A note on derivations, Canad. Math. Bull., 21(3), 369-370, 1978.
  • I. N. Herstein, A note on derivations II, Canad. Math. Bull., 22(4), 509-511, 1979.
  • I. N. Herstein, Topics in Ring Theory, The Univ. of Chicago Press, 1969.
  • H. Kandamar, The k-Derivation of a Gamma-Ring, Turk. J. Math., 23(3), 221-229, 2000.
  • A. R. Khan, M. A. Chaudhry, I. Javaid, Generalized Derivations on Prime Γ-Rings, World Appl. Sci. J., 23(12), 59-64, 2013.
  • S. Kyuno, Gamma Rings, Hadronic Press, 1991.
  • N. Nobusawa, On a generalization of the ring theory, Osaka J. Math., 1, 81-89, 1964.
  • E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8, 1093-1100, 1957.
  • N. N. Suliman, A. H. Majeed, Lie Ideals in Prime Γ-Rings with Derivations, Discussiones Mathe
  • maticae, 33, 49-56, 2013.
There are 13 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Okan Arslan This is me

Hatice Kandamar This is me

Publication Date January 22, 2015
Published in Issue Year 2015 Volume: 2 Issue: 1

Cite

APA Arslan, O., & Kandamar, H. (2015). γ-Lie structures in γ-prime gamma rings with derivations. Journal of Algebra Combinatorics Discrete Structures and Applications, 2(1), 25-37. https://doi.org/10.13069/jacodesmath.87481
AMA Arslan O, Kandamar H. γ-Lie structures in γ-prime gamma rings with derivations. Journal of Algebra Combinatorics Discrete Structures and Applications. March 2015;2(1):25-37. doi:10.13069/jacodesmath.87481
Chicago Arslan, Okan, and Hatice Kandamar. “γ-Lie Structures in γ-Prime Gamma Rings With Derivations”. Journal of Algebra Combinatorics Discrete Structures and Applications 2, no. 1 (March 2015): 25-37. https://doi.org/10.13069/jacodesmath.87481.
EndNote Arslan O, Kandamar H (March 1, 2015) γ-Lie structures in γ-prime gamma rings with derivations. Journal of Algebra Combinatorics Discrete Structures and Applications 2 1 25–37.
IEEE O. Arslan and H. Kandamar, “γ-Lie structures in γ-prime gamma rings with derivations”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 2, no. 1, pp. 25–37, 2015, doi: 10.13069/jacodesmath.87481.
ISNAD Arslan, Okan - Kandamar, Hatice. “γ-Lie Structures in γ-Prime Gamma Rings With Derivations”. Journal of Algebra Combinatorics Discrete Structures and Applications 2/1 (March 2015), 25-37. https://doi.org/10.13069/jacodesmath.87481.
JAMA Arslan O, Kandamar H. γ-Lie structures in γ-prime gamma rings with derivations. Journal of Algebra Combinatorics Discrete Structures and Applications. 2015;2:25–37.
MLA Arslan, Okan and Hatice Kandamar. “γ-Lie Structures in γ-Prime Gamma Rings With Derivations”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 2, no. 1, 2015, pp. 25-37, doi:10.13069/jacodesmath.87481.
Vancouver Arslan O, Kandamar H. γ-Lie structures in γ-prime gamma rings with derivations. Journal of Algebra Combinatorics Discrete Structures and Applications. 2015;2(1):25-37.