All near-rings are 0-symmetric and right distributive. A \(\Gamma\)-near- ring \((M, +, \Gamma)\) is a set \(M\) and a set of binary operators \(\Gamma\) on \(M\) such that \((M, +, \gamma)\) is a near-ring for each \(\gamma \in \Gamma\), and a generalized associative law holds. A \(\sigma\)-subnear-ring of a near-ring \(N\) is a subnear-ring \(N\) which has a special generating set and the idea generalizes that of invariant subnear-ring. This leads to the idea of \(\sigma\)-hereditary classes of near-rings and \(\sigma\)- special classes and radicals. The authors investigate these ideas for near-rings and \(\Gamma\)-near- rings. After establishing basic properties of these concepts they show that three radicals are \(\sigma\)-special: the equiprime radical, the strongly equiprime radical and the \(J_3\)-radical. An example of a special radical which is not \(\sigma\)-special is given. The \(\sigma\)- special radicals of near-rings have very strong hereditary properties.