One-sided ideals in near-rings of transformations

Abstract

Let (G, +) be an arbitrary group and let T0(G) = {f∈ Map(G, G): 0f = 0}; the system composed of T0(G) and the operations of pointwise addition and composition of functions form a (left) near-ring. Berman and Silverman, in their investigation of near-rings of transformations [3], found that for every group G the associated near-ring of transformations T0(G) has no proper ideals. In the present paper left and right ideals of T0(G) are considered.

The endomorphism near-rings of the symmetric groups of degree at least five

“…near-ring is a normal subgroup closed under right multiplication. So the rest follows from Theorem 2.4 and a description of the right ideals of M 0 (H) obtainable from Heatherly (1972) or Pilz (1977). COROLLARY 3.4.…”

“…Using the results of Heatherly (1972) or Pilz (1977), we can write M 0 {H) as a direct sum of right ideals as follows :…”

Maximal right ideals of transformation near-rings

“…Transformation near-rings 411 PROOF. Heatherly (1972) has shown that if x is a nonzero element of G, then v4({x}) is a maximal right ideal of T 0 (G). Now suppose S is a maximal element of 3b.…”

“…The purpose of this note is to characterize maximal right ideals of T 0 (G), the (left) mear-ring of transformations from a group (G, + ) into itself which leave zero fixed. Using this characterization, we answer some questions raised by Heatherly (1972).…”

“…e S. Hence, S e 82 and since S is a maximal right ideal, then S must be a maximal element of ^. Heatherly (1972) considered the right ideal D = {aeT 0 (G)| | support a | < | G | } , where support a = {x e G | (x)ce # 0} and where G is an infinite group. He raised the question: Is D a maximal right ideal of T O (G)1 Using Theorem 3, it is easy to see that the answer is no.…”