It is known that many special radicalsRsatisfy the matrix conditionan arbitrary ring, but that these radicals do not carry over from the base ringRto structural matrix rings in this natural way. In this paper we study the radicals (of structural matrix rings) determined by some polynomial regularities. We obtain various classes of polynomial regularities and classes of structural matrix rings for which the corresponding radicals behave differently from special radicals. For example, on the one hand, some radicals carry over to certain structural matrix rings in the mentioned natural way, whereas on the other hand it might happen that the radical of a structural matrix ring, which is not a complete matrix ring, is {0}, in spite of the fact that sucha radical satisfies the matrix condition, for example the Von Neumann radical. It turns out that the presence or absence of reflexivity and antisymmetry of the underlying Boolean matrix of a structural matrix ring plays a crucial role in this regard.