Near-rings of continuous functions on disconnected groups

Abstract

N(G) denotes the near-ring of all continuous selfmaps of the topological group G (under composition and the pointwise induced operation) and N0(G) is the subnear-ring of N(G) consisting of all functions having the identity element of G fixed. It is known that if G is discrete then (a) N0(G) is simple and (b) N(G) is simple if and only if G is not of order 2. We begin a study of the ideal structure of these near-rings when G is a disconnected group.

Strongly Prime Ideals of Near-rings of Continuous Functions

“…, which may be nontrivial [5]. Hence N 0 (G) need not be simple, in contrast to the situation for the near-ring M 0 (G) of all zeropreserving self-maps of G. Investigations of strongly prime ideals in near-rings of continuous functions commenced in [2], and continued in [1].…”

Near-rings of homotopy classes of continuous functions

“…is an isomorphism of abelian groups. Since the right diagram is commutative, we have for all / G Hom(G, G) and c+G a e TT 0 G: [6] it is not difficult to show that for a locally compact abelian group G with more than two connected components the near-ring [G, G] is not an abstract affine near-ring. Therefore, these near-rings must be investigated in another way.…”

Near-rings of mappings on finite topological groups

“…That is, G has a topology if and only if H is a normal subgroup such that a basis for the open sets of G consists of the cosets of H (see for example [2]). Then G is disconnected, the connected component of 0 being H. Now, if G is an infinite Hausdorff group and C is the connected component of 0, Hofer [3] defined M o = {/ e N 0 (G)\f~\0) contains a clopen set about 0}, P = P(C) = { / G N(G)|range of / c C} and P o = P C\ N o and observed that P is an ideal in N(G) and P o and M o are ideals in N Q (G) such that M 0 \P 0 # 0 . In our case, although G is not T 2 , these results are still true.…”

“…For general results on near-rings, the reader is referred to Pilz [8] and in this paper all near-rings will be right near-rings. Unless otherwise stated G will denote a finite group, and in the next section we will apply some ideas of Hofer [3] to obtain information about the (left) ideals in N(G) and N 0 (G). In the third section we look at two subnear-rings of N 0 (G) determined by [2] Near-rings of mappings 93 endomorphisms, namely the intersection of N 0 (G) with the endomorphism nearring E(G), and the near-ring distributively generated by continuous elements of E(G).…”