On commutativity of prime near-rings with multiplicative generalized derivation

In the present paper, we shall prove that 3-prime near-ring N is commutative ring, if any one of the following conditions are satisfied: (i) f(N)⊆Z, (ii) f([x,y])=0, (iii) f([x,y])=±[x,y], (iv) f([x,y])=±(xoy), (v) f([x,y])=[f(x),y], (vi) f([x,y])=[x,f(y)], (vii) f([x,y])=[d(x),y], (viii) f([x,y])=d(x)oy,(ix) [f(x),y]∈Z for all x,y∈N where f is a nonzero multiplicative generalized derivation of N associated with a multiplicative derivation d.

Keywords:

Prime near-ring derivation, multiplicative generalized derivation,

PDF

___

Bell, H. and Mason, G., On derivations in near rings, Near rings and Near fields, North-Holland Mathematical Studies, 137, (1987), 31-35.
Bell, H. E., On derivations in near-rings II, Kluwer Academic Pub. Math. Appl., Dordr., 426, (1997), 191-197.
Daif, M. N., When is a multiplicative derivation additive, Int. J. Math. Math. Sci., 14 (3), (1991), 615-618.
Daif, M. N. , Bell, H. E., Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci., 15(1), (1992), 205-206.
Daif, M. N. and Tammam El-Sayiad, M. S., Multiplicative generalized derivations which are additive, East-west J. Math., 9 (1), (2007), 31-37.
Bedir, Z., Gölbaşı, Ö., Notes on prime near rings with multiplicative derivation, Cumhuriyet University Faculty of Science, Science Journal (CSJ), (2017), Vol. 38, No. 2, 355-363.
Goldman, H. and Semrl, P., Multiplicative derivations on C(X), Monatsh Math., 121 (3), (1969), 189-197.
Kamal, A. M. and Al-Shaalan, K. H., Existence of derivations on near-rings, Math. Slovaca, 63, (2013), No:3, 431-438.
Martindale III, W. S., When are multiplicative maps additive, Proc. Amer. Math. Soc., 21, (1969), 695-698.
Pilz, G., Near-rings, 2nd Ed. North Holland, Amsterdam, (1983).

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics-Cover

ISSN: 1303-5991
Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 1948
Yayıncı: Ankara Üniversitesi

Arşiv