A distributive near ring is an associative ring with not necessarily commutative addition. In the variety \(\mathcal W\) of distributive near-rings, if a radical class \(\mathcal R\) contains all \(A\in {\mathcal W}\) with \(A^ 2 = 0\), or \(\mathcal R\) is invariantly strong (that is, \({\mathcal R}(A)\) contains all invariant subgroups \(S\) with \(S\in {\mathcal R}\)), then \(\mathcal R\) has the ADS-property (that is, \(I\triangleleft A\in {\mathcal W}\) implies \({\mathcal R}(I)\triangleleft A\)). The semisimple class \({\mathcal S}{\mathcal R}\) of a radical \(\mathcal R\) is hereditary if and only if \({\mathcal R}(I)A+A{\mathcal R}(A)\subseteq {\mathcal R}(A)\) for all \(I\triangleleft A\in {\mathcal W}\). If \({\mathcal S}{\mathcal R}\) consists of rings, then \(R\) is invariantly strong. Also it is shown that not all semisimple classes in \({\mathcal W}\) are hereditary.