In a previous work , the current author, together with P. R. Hall, studied various kinds of primeness in near-rings and sandwich near-rings of continuous functions. In this paper, we study various prime radicals for sandwich near-rings. In certain cases, complete characterisations of the prime, 3-prime, equiprime and strongly equiprime radicals are obtained.
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.
We define a prime ΓM-module for a Γ-ringM. It is shown that a subsetP ofM is a prime ideal ofM if and only ifP is the annihilator of some prime ΓM-moduleG. s-prime ideals ofM were defined by the first author. We defines-modules ofM, analogous to a concept defined by De Wet for rings. It is shown that a subsetQ ofM is ans-prime ideal ofM if and only ifQ is the annihilator of somes-moduleG ofM. Relationships between prime ΓM-modules and primeR-modules are established, whereR is the right operator ring ofM. Similar results are obtained fors-modules.
A radical α in the universal class of associative rings is called matric-extensible if α (Rn) = (α (R))n for any ring R, and natural number n, where Rn denotes the nxn matrix ring with entries from R. We investigate matric-extensibility of the lower radical determined by a simple ring S. This enables us to find necessary and sufficient conditions for the lower radical determined by S to be an atom in the lattice of hereditary matric-extensible radicals. We also show that this lattice has atoms which are
not of this form. We then describe all atoms of the lattice, and show that it is atomic.
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.
We continue our study of the lattice of matric-extensible radicals of associative rings. We find some atoms generated by
simple rings of the lattices of all matric-extensible radicals, matric-extensible supernilpotent radicals and matric-extensible
special radicals. We consider *-rings, which were previously defined by the second author, and consider when they generate
atoms of these lattices.
Authors:G.F. Birkenmeier, G.L. Booth and N.J. Groenewald
A class K of rings has the GADS property (i.e., generalized ADS property) if wheneverX& I& R with X∈ K, then there exists B & R with B ∈ K such that X ⊆ B ⊆ I. Radicals whose semisimple classes have the GADS property are called g-radicals. In this paper, we fully characterize the class of g -radicals. We show that ? is a g-radical if and only if either ? ⊆ I or S? ⊆ I , whereI denotes the class of idempotent rings and S? denotes the semisimple class of ?. It is also shown that the (hereditary) g-radicals form an (atomic) sublattice of the lattice of all radicals.