The Bicentralizer Nearrings of R

Abstract

Let Γ be an additive (not necessarily abelian) topological group. As is customary, an automorphism of Γ will be assumed to be a homeomorphism as well and endomorphisms will be assumed to be continuous. For any group G, of automorphisms of Γ, we will denote by N _G (Γ) the bicentralizer nearring of Γ, with respect to G, which consists of all continuous selfmaps of F which commute with all elements of G. In this paper, we take Γ = R, the additive group of real numbers and we determine all the bicentralizer nearrings of R. It is well known that any automorphism t of R is of the form t(x) = ax for some a ≠ 0 and we will denote this map by t _a We will denote by Aut(R) and End(R), the automorphism group and the endomorphism ring, respectively, of R. Finally, it is known that the only endomorphism of R which is not an automorphism is the map which sends everything into zero so that End(R) = {t₀} ⋃ Aut(R). Moreover, the mapping which sends a into t _a is an isomorphism from R _? , the multiplicative group of nonzero real numbers, onto Aut(R). We will adhere to the convention that if G is any subgroup of R _? , then [_G] is the subgroup of Aut(R) which corresponds to G. That is [G] = {t_a : a ∈ G}. For our first result, we choose the smallest (with respect to cardinality) nontrivial subgroup of R _? Its proof consists of one line and even that is probably not necessary.