Let \({\mathcal A}\) be a universal class of rings or near rings (not necessarily associative). The authors show that the Brown-McCoy radical class \({\mathcal G}\) in \({\mathcal A}\) coincides with the uniquely determined largest homomorphically closed class of rings (near rings) in \({\mathcal A}\) without unity. Besides this characterization as a lower radical they present also a characterization of \({\mathcal G}\) as an upper radical in a universal class \({\mathcal A}\) of alternative or near rings. In fact, \({\mathcal G}\) is the upper radical determined by the uniquely determined largest hereditary class of rings (near rings) in \({\mathcal A}\) with unity. With respect to this upper class representation, \({\mathcal G}\) has the intersection property. For associative rings A, it is shown that \(r_ 3(A)=A\) in A-mod if and only if \({\mathcal G}(A)=A\) in the category of rings, where \(r_ 3\) is a module-radical, introduced by the first author [Publ. Math. 27, 7-12 (1980; Zbl 0456.16009)].