Abstract
AbstractWe establish sufficient conditions for 3-prime near-rings to be commutative rings. In particular, for a 3-prime near-ring R with a derivation d, we investigate conditions such as $$d([U,V])\subseteq Z(R)$$
d
(
[
U
,
V
]
)
⊆
Z
(
R
)
, $$d(U)\subseteq Z(R)$$
d
(
U
)
⊆
Z
(
R
)
, $$x_{o}d(R)\subseteq Z(R)$$
x
o
d
(
R
)
⊆
Z
(
R
)
, and $$Ux_{o}\subseteq Z(R)$$
U
x
o
⊆
Z
(
R
)
. As a by-product, we generalize and extend known results related to rings and near-rings. Furthermore, we discuss the converse of a well-known result in rings and near-rings, namely: if $$x\in Z(R)$$
x
∈
Z
(
R
)
, then $$d(x)\in Z(R)$$
d
(
x
)
∈
Z
(
R
)
. In addition, we provide useful examples illustrating our results.
Publisher
Springer Science and Business Media LLC