Let \(R\) be a left near-ring. We denote the prime and completely prime radicals of \(R\) by \({\mathbf P}_ 0(R)\) and \({\mathbf P}_ 2(R)\) respectively. In this paper, the concept of a 2-primal ideal of a near- ring is introduced. Let \(N(R)\) denote the set of nilpotent elements of \(R\). An ideal \(I\) of \(R\) is called 2-primal if \({\mathbf P}_ 0 (R/I) = N(R/I)\). \(R\) is 2-primal if the zero ideal is 2-primal. It is shown that if \(I\) is an ideal of \(R\), then \(I\) is a completely semiprime ideal of \(R\) if and only if \(I\) is both semiprime and 2-primal. If \(IR \subseteq I\), then the following conditions are equivalent: (i) \(I\) is a completely prime ideal; (ii) \(I\) is a prime and a completely semiprime ideal; (iii) \(I\) is a prime and a 2-primal ideal. If \(R\) is zero-symmetric, the following conditions are equivalent: (i) \(R\) is 2-primal; (ii) if \(P\) is a prime ideal of \(R\) and \(A_ 1, \dots ,A_ n\) are nonempty subsets of \(R\) such that \(A_ 1 \dots A_ n = 0\) then \(A_ i \subseteq P\) for some \(i\); (iii) \({\mathbf P}_ 0(R) = {\mathbf P}_ 2(R)\). The paper then considers the radicals \({\mathbf P}_ 0\) and \({\mathbf P}_ 2\), and their relation to 2-primal near-rings. It is shown that if \(S\) is a subnear- ring of a 2-primal near-ring, then \(S\) is 2-primal and \({\mathbf P}_ 0(S) = S \cap {\mathbf P}_ 0(R)\). Moreover \({\mathbf P}_ 2\) is a radical map and \({\mathbf P}_ 2(S) \subseteq S \cap {\mathbf P}_ 2(R)\) for a subring \(S\) of a near-ring \(R\). Equality holds if \(S\) is a direct summand. Denote by \({\mathcal R}^ 2\) (\({\mathcal R}_ 0^ 2\)) the class of (zero-symmetric) near-rings for which every prime ideal is completely prime. Among the results proved here is: If \(R\) is a zero-symmetric near-ring, then \(R \in {\mathcal R}^ 2_ 0\) if and only if every ideal of \(R\) is 2-primal. If \(R \in {\mathcal R}^ 2_ 0\) and \(I\) is an ideal of \(R\) such that every prime ideal of \(I\) is an ideal of \(R\) then \(I \in {\mathcal R}^ 2_ 0\). Certain results are also given for the class of near-rings for which every semiprime ideal is completely semiprime. Various examples are given to illustrate the theory.