On base radical and semisimple classes defined by class operators

On Base Radical Operators for Classes of Finite Associative Rings

“…When every homomorphic image of each accessible subring of A ∈ A is isomorphic to an accessible subring of a homomorphic image of A, the dual of Theorem 2.2(iv) holds and S¬U(X) = U¬S(X) [9].…”

“…The proof arguments are said to be dualised and are only sometimes included. The base radical operator US and base semisimple operator SU are then US(X) = {A ∈ A | every nonzero homomorphic image of A has a nonzero accessible subring in X} and SU(X) = {A ∈ A | every nonzero accessible subring of A has a nonzero homomorphic image in X} [9]. A class X ⊆ A is a base radical class if and only if X = US(X) and, dually, X is a base semisimple class if and only if X = SU(X) [10].…”

“…The notion of generating an operator semigroup through repeated application of the U and S operators on all subclasses X ⊆ A was introduced in [9] and some general properties of this semigroup and its possible orders in a more general setting, which includes finite associative rings, were detailed in [14]. The semigroup, denoted RT A with order |RT A |, contains all distinct operators composed of U and S which can act on subclasses of a universal class A.…”