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 Primeness and Primitivity in Near-rings J. Korean Math. Soc.Published online December 15, 2020 Gerhard Wendt JKU Abstract : 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 Full-Text :