( σ , τ ) -derivations on prime near rings

Mohammad Ashraf; Asma Ali; Shakir Ali

Archivum Mathematicum (2004)

  • Volume: 040, Issue: 3, page 281-286
  • ISSN: 0044-8753

Abstract

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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 ( σ , τ ) - derivation where σ , τ are automorphisms of the near-ring. Finally, it is shown that under appropriate additional hypothesis a near-ring must be a commutative ring.

How to cite

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Ashraf, Mohammad, Ali, Asma, and Ali, Shakir. "$(\sigma ,\tau )$-derivations on prime near rings." Archivum Mathematicum 040.3 (2004): 281-286. <http://eudml.org/doc/249295>.

@article{Ashraf2004,
abstract = {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.},
author = {Ashraf, Mohammad, Ali, Asma, Ali, Shakir},
journal = {Archivum Mathematicum},
keywords = {prime near-ring; derivation; $\sigma $-derivation; $(\sigma , \tau )$-derivation; $(\sigma , \tau )$-commuting derivation; prime near-rings; commuting derivations; commutativity theorems},
language = {eng},
number = {3},
pages = {281-286},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {$(\sigma ,\tau )$-derivations on prime near rings},
url = {http://eudml.org/doc/249295},
volume = {040},
year = {2004},
}

TY - JOUR
AU - Ashraf, Mohammad
AU - Ali, Asma
AU - Ali, Shakir
TI - $(\sigma ,\tau )$-derivations on prime near rings
JO - Archivum Mathematicum
PY - 2004
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 040
IS - 3
SP - 281
EP - 286
AB - 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.
LA - eng
KW - prime near-ring; derivation; $\sigma $-derivation; $(\sigma , \tau )$-derivation; $(\sigma , \tau )$-commuting derivation; prime near-rings; commuting derivations; commutativity theorems
UR - http://eudml.org/doc/249295
ER -

References

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  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. (1996) Zbl0849.16039MR1386483
  2. Bell H. E., On derivations in near-rings, II, Kluwer Academic Publishers Netherlands (1997), 191–197. (1997) Zbl0911.16026MR1492193
  3. Bell H. E., Mason G., On derivations in near-rings and rings, Math. J. Okayama Univ. 34 (1992), 135–144. (1992) Zbl0810.16042MR1272613
  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. (1987) Zbl0619.16024MR0890753
  5. Kamal Ahmad A. M., σ - derivations on prime near-rings, Tamkang J. Math. 32 2 (2001), 89–93. MR1826415
  6. Meldrum J. D. P., Near-rings and Their Link with Groups, Pitman, 1985. (1985) MR0854275
  7. Posner E. C., Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100. (1957) MR0095863
  8. Wang X. K., Derivations in prime near-rings, Proc. Amer. Math. Soc. 121 (1994), 361–366. (1994) Zbl0811.16040MR1181177

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