Abstract
We 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
]
)
⊆
Z
(
R
)
,
d
(
U
)
⊆
Z
(
R
)
,
x
o
d
(
R
)
⊆
Z
(
R
)
, and
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
∈
Z
(
R
)
, then
d
(
x
)
∈
Z
(
R
)
. In addition, we provide useful examples illustrating our results.