BibTex RIS Kaynak Göster

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

Yıl 2015, , 25 - 37, 22.01.2015
https://doi.org/10.13069/jacodesmath.87481

Öz

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$M=2$.

Kaynakça

  • 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

Yıl 2015, , 25 - 37, 22.01.2015
https://doi.org/10.13069/jacodesmath.87481

Öz

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$M=2$.

Kaynakça

  • 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.
Toplam 13 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Makaleler
Yazarlar

Okan Arslan Bu kişi benim

Hatice Kandamar Bu kişi benim

Yayımlanma Tarihi 22 Ocak 2015
Yayımlandığı Sayı Yıl 2015

Kaynak Göster

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. Mart 2015;2(1):25-37. doi:10.13069/jacodesmath.87481
Chicago Arslan, Okan, ve Hatice Kandamar. “γ-Lie Structures in γ-Prime Gamma Rings With Derivations”. Journal of Algebra Combinatorics Discrete Structures and Applications 2, sy. 1 (Mart 2015): 25-37. https://doi.org/10.13069/jacodesmath.87481.
EndNote Arslan O, Kandamar H (01 Mart 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 ve H. Kandamar, “γ-Lie structures in γ-prime gamma rings with derivations”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 2, sy. 1, ss. 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 (Mart 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 ve Hatice Kandamar. “γ-Lie Structures in γ-Prime Gamma Rings With Derivations”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 2, sy. 1, 2015, ss. 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.