ICMC 2017: Mathematics and Computing pp 380-390

# Derivations on Lie Ideals of Prime $$\varGamma$$-Rings

• Kalyan Kumar Dey
• Akhil Chandra Paul
• Bijan Davvaz
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 655)

## Abstract

Let M be a 2-torsion free prime $${\varGamma }$$-ring with Z(M) as the center of M. In this paper, we prove the following: (i) If U is a Lie ideal of M and if $$d\not =0$$ is a derivation of M such that $$d^2(U) = 0$$, then $$U\subseteq Z(M)$$; (ii) if $$U\not \subset Z(M)$$ is a Lie ideal of M and $$d\not =0$$ is a derivation of M, then $$Z(d(U))\subseteq Z(M)$$; (iii) If $$U\not \subset Z(M)$$ is a Lie ideal of M and if d is a derivation of M such that $$d^3\not = 0$$, then $$d(U)^*$$, the subring generated by d(U) contains a non-zero ideal of M. Finally, we prove that if $$U \not \subset Z(M)$$ is a Lie ideal of M and $$d_1$$ and $$d_2$$ are derivations of M such that $$d_1d_2(U) = 0$$, then $$d_1 = 0$$ or $$d_2 = 0$$.

## Keywords

$$\varGamma$$-ring Prime $$\varGamma$$-ring Derivation $${\varGamma }$$-Lie ideal

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© Springer Nature Singapore Pte Ltd. 2017

## Authors and Affiliations

• Kalyan Kumar Dey
• 1
Email author
• Akhil Chandra Paul
• 1
• Bijan Davvaz
• 2