In the present paper, we 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]) = ±τ ([x, y]), (iv) f ([x, y]) = ±τ(xoy), (v) f ([x, y]) = τ ([d(x), y]), for all x, yN, where f is a nonzero left multiplicative generalized (σ, τ)-derivation of N associated with a multiplicative (σ, τ)-derivation d.

1.
H.
Bell
and
G.
Mason
,
North-Holland Mathematical Studies
,
137
,
31
35
, (
1987
).
2.
H. E.
Bell
,
Math. Appl., Kluwer Acad. Publ., Dordrecht
,
426
,
191
197
, (
1997
).
3.
M. N.
Daif
,
Int. J. Math. Math. Sci.
,
14
(
3
),
615
618
, (
1991
).
4.
M. N.
Daif
, and
M. S. Tammam
El-Sayiad
,
East-west J. Math.
,
9
(
1
),
31
37
, (
2007
).
5.
C.
Hou
,
W.
Zhang
and
Q.
Meng
,
Linear Algebra and Appl.
432
,
2600
2607
(
2010
).
6.
A. M.
Kamal
, and
K. H.
Al-Shaalan
,
Math. Slovaca
,
63
(
3
),
431
438
, (
2013
).
7.
W. S.
Martindale
 III
,
Proc. Amer. Math. Soc.
,
21
,
695
698
, (
1969
).
8.
G.
Pilz
,
Near-rings
, (
North Holland
,
Amsterdam
,
1983
), p.
470
.
9.
E. C.
Posner
,
Proc. Amer. Math. Soc.
,
8
,
1093
1100
, (
1957
).
This content is only available via PDF.
You do not currently have access to this content.