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prime near-ring; derivation; $\sigma $-derivation; $(\sigma, \tau )$-derivation; $(\sigma, \tau )$-commuting derivation
There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation namely $(\sigma ,\tau )$- derivation where $\sigma ,\tau $ are automorphisms of the near-ring. Finally, it is shown that under appropriate additional hypothesis a near-ring must be a commutative ring.
[1] Beidar K. I., Fong Y., Wang X. K.: Posner and Herstein theorems for derivations of 3-prime near-rings. Comm. Algebra 24 (5) (1996), 1581–1589. MR 1386483 | Zbl 0849.16039
[2] Bell H. E.: On derivations in near-rings, II. Kluwer Academic Publishers Netherlands (1997), 191–197. MR 1492193 | Zbl 0911.16026
[3] Bell H. E., Mason G.: On derivations in near-rings and rings. Math. J. Okayama Univ. 34 (1992), 135–144. MR 1272613 | Zbl 0810.16042
[4] Bell H. E., Mason G.: On derivations in near-rings. Near-Rings and Near-Fields (G. Betsch, ed.) North-Holland, Amsterdam (1987), 31–35. MR 0890753 | Zbl 0619.16024
[5] Kamal Ahmad A. M.: $\sigma $- derivations on prime near-rings. Tamkang J. Math. 32 2 (2001), 89–93. MR 1826415
[6] Meldrum J. D. P.: Near-rings and Their Link with Groups. Pitman, 1985. MR 0854275
[7] Posner E. C.: Derivations in prime rings. Proc. Amer. Math. Soc. 8 (1957), 1093–1100. MR 0095863
[8] Wang X. K.: Derivations in prime near-rings. Proc. Amer. Math. Soc. 121 (1994), 361–366. MR 1181177 | Zbl 0811.16040
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