On the commutativity conditions for rings and $\Gamma$-rings
On the commutativity conditions for rings and $\Gamma$-rings
Let $R$ be any ring. In this paper we observe the relation between the center of $R$-ring $R$ and the center of usual ring $R$ and then prove if the center of $R$-ring $R$ is nonzero, then $R$ is commutative as a ring. We also show that the common hypothesis\[a\alpha b\beta c=a\beta b\alpha c\: \text{for all}\: a,b,c\in M\:\text{and}\: \alpha,\beta\in\Gamma\]for a weak Nobusawa $\Gamma$-ring $M$ is sufficent for $M$ to be commutative. Also, we investigate some conditions on ideals of $\Gamma$-ring that make $M$ to be commutative.
Keywords:
prime ring, commutativity prime $\Gamma$-ring, ideal,
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- [1] W.E. Barnes, On the Γ-rings of Nobusawa, Pacific J. Math. 18 (3), 411–422, 1966.
- [2] W.E. Coppage and J. Luh, Radicals of gamma rings, J. Math. Soc. Japan, 23 (1), 40–52, 1971.
- [3] K.K. Dey and A.C. Paul, Free Actions on Semiprime Gamma Rings, Gen. Math. Notes, 5 (1), 7–14, 2011.
- [4] Md.F. Hoque and A.C. Paul, On Centralizers of Semiprime Gamma Rings, Int. Math. Forum, 6 (13), 627–638, 2011.
- [5] S. Kyuno, On the radicals of gamma rings, Osaka J. Math. 12, 639–645, 1975.
- [6] S. Kyuno, Gamma Rings, Hadronic Press, 1991.
- [7] N. Nobusawa, On a generalization of the ring theory, Osaka J. Math. 1, 81–89, 1964.
- Yayın Aralığı: Yılda 6 Sayı
- Başlangıç: 2002
- Yayıncı: Hacettepe Üniversitesi Fen Fakultesi
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