Journal of the
Korean Mathematical Society JKMS

In near-ring theory several different types of primitivity exist. These all imply several different types of primeness. In case of near-rings with DCCN most of the types of primeness are known to imply primitivity of a certain kind. We are able to show that also so called $1$-prime near-rings imply $1$-primitivity. This enables us to classify maximal ideals in near-rings with chain condition with the concept of $1$-primeness which leads to further results in the structure theory of near-rings.

Keywords: $1$-prime near-rings, $1$-primitive near-rings, maximal ideals, simple near-rings

MSC numbers: 16Y30