Abstract
Let N denote a 3–prime zero-symmetric left near-ring, and let d be a nonzero derivation on N. Let U ≠ φ be a nonzero subset of N such that (i) U N ⊆ U or (ii) N U ⊆ U. We prove that N must be a commutative ring if one of the following holds: (a) U satisfies one of (i) and (ii), and d(U) is multiplicatively central; (b) U satisfies both of (i) and (ii), d 2 ≠ 0, and [d(U), d(U)] = {0}. Some related results are also given.
Supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. 3961
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References
Beidar, K. I., Fong, Y., and Wang, X. K., Posner and Herstein theorems for derivations of 3-prime near-rings, to appear.
Bell, H. E. and Mason, G. (1987), On derivations in near-rings, in: Near-rings and Near-fields (G. Betsch, ed.), North-Holland, Amsterdam, 31–35.
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Wang, X. K. (1994), Derivations in prime near-rings, Proc. Amer. Math. Soc. 121, 361–366.
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© 1997 Kluwer Academic Publishers
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Bell, H.E. (1997). On Derivations in Near-Rings, II. In: Saad, G., Thomsen, M.J. (eds) Nearrings, Nearfields and K-Loops. Mathematics and Its Applications, vol 426. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1481-0_10
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DOI: https://doi.org/10.1007/978-94-009-1481-0_10
Publisher Name: Springer, Dordrecht
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