Group Homomorphism Generated Near-Rings and Rings: A Unity Not Fixing Each Element of the Group

Abstract

Let (G, +) be a group, not necessarily abelian, and let K be a nontrivial subgroup of G. Let ℋ =  ℋ(G, K) be the additive group generated by Hom (G, K). Then (ℋ(G, K), +, ○) is a d.g. near-ring. If K ≠ G, then ℋ(G, K) cannot contain the unity element of ℰ(G), the near-ring generated by End G. Surprisingly, examples exist which show it may indeed have a two-sided unity element. Conditions are developed involving G and K¯, the additive subgroup generated by ∪ {h(G) : h ∈ ℋ}, which characterize when ℋ(G, K) contains a one-sided or two-sided unity element. The cases when K¯ is abelian or an E-group are considered. As a consequence of this theory, connections between ℰ(G) and ℰ(K¯), via ℋ(G, K), are established. Numerous illustrative examples are given.

Acknowledgments

This work is part of the author's doctoral dissertation at the University of Louisiana at Lafayette under Professors G. F. Birkenmeier and H. E. Heatherly. The author wishes to thank them for their helpful suggestions, comments, criticisms, and most of all their time and willingness to allow him to roam relatively freely down paths of research opened by them in Birkenmeier et al. (1997a,b). The author also thanks the referees for their careful reading of an earlier version, resulting in an improved version of the paper.