The base radical class Lb(X), generated by a class X was introduced in . It consists of those rings whose nonzero homomorphic images have nonzero accessible subrings in X. When X is homomorphically closed, Lb(X) is the lower radical class defined by X, but otherwise X may not be contained in Lb(X). We prove that for a hereditary radical class L with semisimple class S(R), Lb(S(R)) is the class of strongly R-semisimple rings if and only if R is supernilpotent or subidempotent. A number of further examples of radical classes of the form Lb(X) are discussed.
It is shown that in a well known characterization of radical classes, closure under unions of chains of ideals can be replaced
by closure under unions of continuous well-ordered chains of ideals. Some consequences are discussed.
We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which
applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and
element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties
are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular
varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is
subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second
section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based
on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in . We consider the subtractive
case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of
the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined
by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal
varieties of algebras, most of which are subtractive.
The Brown-McCoy radical
is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals,
, respectively, for zerosymmetric Γ-near-rings. Both
are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric Γ-near-ring with left operator near-ringL, it is shown that
, with equality ifM has a strong left unity.
is extended to the variety of arbitrary near-rings, and
are extended to the variety of arbitrary Γ-near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA ⊲N, then
, with equality ifA if left invariant.
It was previously shown that every special radical classR of rings induces a special radical class ?R of G-rings. Amongst the special radical classes of near-rings, there are some, called the s-special radical classes, which induce, special radical classes of G-near-rings by the same procedure as used in the ring case. The s-special radicals of near-rings possess very strong hereditary properties. In particular, this leads to some new results for the equiprime andI3 radicals.