Research Article Associate, Hyperdomainlike, and Presimplifiable Hyperrings Agboola Adesina Abdul Akeem 1 and Davvaz Bijan 2 1 Department of Mathematics, Federal University of Agriculture, PMB 2240, Abeokuta, Ogun State, Nigeria 2 Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran Correspondence should be addressed to Agboola Adesina Abdul Akeem; aaaola2003@yahoo.com Received 12 February 2014; Accepted 17 June 2014; Published 14 August 2014 Academic Editor: Feng Feng Copyright © 2014 A. Adesina Abdul Akeem and D. Bijan. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on the works of Axtell et al., Anderson et al., and Ghanem on associate, domainlike, and presimplifable rings, we introduce new hyperrings called associate, hyperdomainlike, and presimplifable hyperrings. Some elementary properties of these new hyperrings and their relationships are presented. 1. Introduction Te study of strongly associate rings began with Kaplansky in [1] and was further studied in [2–5]. Domainlike rings and their properties were presented by Axtell et al. in [6]. Presimplifable rings were introduced by Bouvier in the series of papers [7–11] and were later studied in [2–4]. Further properties of associate and presimplifable rings were recently presented by Ghanem in [12]. Te theory of hyperstructures was introduced in 1934 by Marty [13] at the 8th Congress of Scandinavian Mathemati- cians. Introduction of the theory has caught the attention and interest of many mathematicians and the theory is now spreading like wild fre. Te notion of canonical hypergroups was introduced by Mittas [14]. Some further contributions to the theory can be found in [15–19]. Hyperrings are essentially rings with approximately mod- ifed axioms. Hyperrings (, +, ⋅) are of diferent types intro- duced by diferent researchers. Krasner [20] introduced a type of hyperring (, +, ⋅) where + is a hyperoperation and ⋅ is an ordinary binary operation. Such a hyperring is called a Kras- ner hyperring. Rota in [21] introduced a type of hyperring (, +, ⋅) where + is an ordinary binary operation and ⋅ is a hyperoperation. Such a hyperring is called a multiplicative hyperring. de Salvo [22] introduced and studied a type of hyperring (, +, ⋅) where + and ⋅ are hyperoperations. Te most comprehensive reference for hyperrings is Davvaz and Leoreanu-Fotea’s book [18]. Some other references are [23– 31]. In this paper, we present and study associate, hyperdo- mainlike, and presimplifable hyperrings. Te relationships between these new hyperrings are presented. 2. Preliminaries In this section, we will provide some defnitions that will be used in the sequel. For full details about associate, domain- like, and presimplifable rings, the reader should see [1, 4– 6, 12]. Also, for details about hyperstructures and hyperrings, the reader should see [12]. Defnition 1. Let  be a commutative ring with unity. (1)  is called an associate ring if whenever any two elements , ∈  generate the same principal ideal of , there is a unit  ∈ () such that  = . (2)  is called a domainlike ring if all zero divisors of  are nilpotent. (3)  is called a presimplifable ring if, for any two elements , ∈  with =, we have =0 or  ∈ (). (4)  is called a superassociate ring if every subring of  is associate. Hindawi Publishing Corporation Journal of Mathematics Volume 2014, Article ID 458603, 7 pages http://dx.doi.org/10.1155/2014/458603