Special radicals of near-rings and Γ-near-rings

Abstract

It was previously shown that every special radical class ~ of rings induces a special radical class p~ of r-rings. Amongst the special radical classes of nearrings, there are some, called the c~-special radical classes, which induce special radical classes of r-near-rings by the same procedure as used in the ring case. The a-special radicals of near-rings possess very strong hereditary properties. In particular, this leads to some new results for the equiprime and 273 radicals. O. IntroductionA most sought after and useful property of a radical ~ in an algebraic variety is the following: If I is an ideal of N E I), then 7~(I) = I N 7~(N). All the wellknown radicals in the variety of associative rings enjoy this property. However, this is not true of many of their generalizations to the variety of 0-symmetric near-rings. Nevertheless, it is well known that the Jacobson type radicals Z2 and 773, and the Brown-McCoy radical exhibit this property. Recently, it has been shown ([10, [11]) that several other radicals, based on the equiprime near-rings, also do so. In the present paper, we will show that many of these radicals satisfy a much stronger conditions, namely that 7~(I) = 1 N ~(N) for any ~,-subnear-ring (and hence any invariant subgroup) 1 of N. This leads us to introduce the concept of ~r-special radicals for 0-symmetric near-rings.In [3] it was shown that every special radical class 7~ of rings induces a special radical class pT~ of F-rings, as follows: Let A/[ be a special class of rings which determines 7~ as an upper radical. Let A74 be the set of all those F-rings M whose left operator ring L is in A//, and such that zrM = 0 implies z = 0, for all x E M. Then .~1 is a special class and pT~ is defined to be the upper radical determined by the class A/[. pT~ is uniquely determined in the sense that if )k41 and A/t2 are two special classes of rings which determine the same upper radical 7~, then the Mathematics subject classification numbers, 1991. Primary 16Y30, 16Y99.

On Jacobson Near-rings and Special Radicals

“…From Lemma 3.12 and [35,Theorem 4.4], we have that R ∈ (SG) c if and only if M n (R) ∈ (SG) c , and R ∈ (SJ 2 ) c if and only if M n (R) ∈ (SJ 2 ) c . It also follows from [5] that R ∈ P e if and only if M n (R) ∈ P e . Hence, (SG) c ∩ P e and (SJ 2 ) c ∩ P e satisfy the matrix extension property.…”

“…The class of 2-primitive near-rings has the matrix extension property [35]. It follows from [5] and [29] that both the class of equiprime near-rings and the class of 0-prime near-rings have the matrix extension property. Recall that for near-rings with unity, 1-, 2-and 3-primitive near-rings coincide.…”

“…We will use the definition given by Veldsman in [37]. Two different meanings assigned to special radicals for near-rings are namely the upper radical determined by a hereditary class which is closed under essential extensions and which consists of 0-prime near-rings [2] or equiprime near-rings [5], respectively.…”

Hereditary right Jacobson radicals of type-1(e) and 2(e) for right near-rings

“…A class E is called hereditary if I ▹ R ∈ E implies I ∈ E. E is called c-hereditary if I is a left invariant ideal of R ∈ E implies I ∈ E. It is clear that a hereditary class is a regular class. If I ▹ R and for every non zero ideal J of R, J ∩ I ̸ = {0}, then I is called an essential ideal of R and is denoted by I ▹• R. A class of near-rings E is called closed under essential extensions (essential left invariant extensions) if I ∈ E, I ▹• R (I is an essential ideal of R which is left invariant) implies R ∈ E. A class of near-rings E is said to satisfy condition (F l ) whenever K ▹ I ▹ R, and I is left invariant in R and I/K ∈ E, it follows that K ▹ R. In [2], G. L. Booth Throughout this section ν ∈ {1, 2}. In this section first we introduce right R-groups of type-ν(e) and study some of their properties.…”

“…Proof. Let g 0 be a generator of the right R-group G. (1) implies (2) follows from the definition of a right R-group of type-ν(e) as g 0 R = G. Assume (2). Suppose that 0 ̸ = g ∈ G, r 1 , r 2 ∈ R and gxr 1 = gxr 2 for all x ∈ R. Since g ̸ = 0 and G is a right R-group of type-ν, gR ̸ = {0} as {h ∈ G | hR = {0}} is an ideal of G. Let < gR > s be the subgroup of (G, +) generated by gR.…”