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.
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$M=2$.
Keywords
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.
Details
| Primary Language | English |
|---|---|
| Journal Section | Articles |
| Authors | |
| Publication Date | January 22, 2015 |
| Published in Issue | Year 2015 Volume: 2 Issue: 1 |