Abstract:
Based on the works of Axtell et al., Anderson et al., and Ghanem on associate, domainlike, and presimplifiable rings, we introduce new hyperrings called associate, hyperdomainlike, and presimplifiable hyperrings. Some elementary properties of these new hyperrings and their relationships are presented. 1. Introduction The 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]. Presimplifiable rings were introduced by Bouvier in the series of papers [7–11] and were later studied in [2–4]. Further properties of associate and presimplifiable rings were recently presented by Ghanem in [12]. The theory of hyperstructures was introduced in 1934 by Marty [13] at the 8th Congress of Scandinavian Mathematicians. Introduction of the theory has caught the attention and interest of many mathematicians and the theory is now spreading like wild fire. The 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 modified axioms. Hyperrings are of different types introduced by different researchers. Krasner [20] introduced a type of hyperring where + is a hyperoperation and is an ordinary binary operation. Such a hyperring is called a Krasner 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. The 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, hyperdomainlike, and presimplifiable hyperrings. The relationships between these new hyperrings are presented. 2. Preliminaries In this section, we will provide some definitions that will be used in the sequel. For full details about associate, domainlike, and presimplifiable rings, the reader should see [1, 4–6, 12]. Also, for details about hyperstructures and hyperrings, the reader should see [12]. Definition 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 presimplifiable ring if, for any two elements with , we have or .(4) is called a superassociate ring if

Abstract:
We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme $(X,\mathcal{O}_X)$ and prove that $\Gamma(X,\mathcal{O}_X)\simeq R$ for any integral affine hyperring scheme $X=Spec R$.

Abstract:
Hyperring is a structure generalizing that of a ring, but where the addition is not a composition, but a hypercomposition, i.e., the sum x+y of two elements, x,y, of a hyperring H is, in general, not an element but a subset of H. When the non-zero elements of a hyperring form a multiplicative group, the hyperring is called a hyperfield, and this structure generalizes that of a field. A certain class of hyperfields (residual hyperfields of valued fields) has been used by the author [1] as an important technical tool in his theory of approximation of complete valued fields by sequences of such fields. Tne non-commutative theory of hyperrings (particularly Artinian) has been studied in depth by Stratigopoulos [2].