A ring means an associative ring, a radical class means an associative ring radical class. For two radical classes \(R_ 1,R_ 2\), let \(R_ 1\vee R_ 2\) be the lower radical class defined by \(R_ 1\cup R_ 2\), and let \(R_ 1\wedge R_ 2 = R_ 1\cap R_ 2\). A radical class \(R\) is small iff \(R\vee T\) is not the class of all rings whenever \(T\) is a radical class which is not the class of all rings. A radical class \(R\) is large iff \(R\wedge T\neq {0}\) for every radical class \(T \neq\{0\}\). The paper contains two general results concerning small and large radical classes (Theorem 1.2 and Proposition 1.10), and a number of corollaries about some special cases.
Let \(R\) be a radical class with semisimple class \(S\). Then the following conditions are equivalent (Theorem 1.2): (i) for every simple ring \(A\) with identity there exists a ring \(B_ A\in S\) such that \(\text{Var}(B_ A) = \text{Var}(A)\), (ii) \(R\) is small. Corollaries: the following radical classes are small: the Brown-McCoy, the lower radical class defined by a single ring, the lower radical class defined by the class of all simple rings. If \(R_ 1\subset R_ 2\) are radical classes, then \(R_ 2\) is small implies \(R_ 1\) is small; hence the Jacobson radical class is small. On the contrary, the generalized nil radical class and the class of all regular rings are not small. For any class \(M\) of rings let \(L(M)\) \((U(M))\) be the lower (the upper) radical class defined by \(M\). A sufficient condition for a radical class \(R\) (with the semi-simple class \(S\)) to be large is given in Proposition 1.10: if every nonzero ring from \(S\) has a nonzero homomorphic image in \(R\), then \(R\) is large. Corollaries: if \(V\) is a ring variety then \(L(V) \vee U(V)\) is large; if \(M\) is the class of all simple rings then \(L(M)\vee U(M)\) is large. Some examples of radical classes which are not large are also given. In Section 3 the authors consider small radical classes for some other algebraic structures. Theorem 3.1 shows that all nontrivial radical classes of groups, abelian groups, nonassociative rings are not small, so the concept of small radical classes is not interesting for such structures.