Cyclic modules over fundamental rings derived from strongly regular equivalences
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Cyclic modules over fundamental rings derived from strongly regular equivalences S. Mirvakili1 · P. Ghiasvand1 · B. Davvaz2
Received: 16 February 2016 / Accepted: 3 October 2016 © Fondation Carl-Herz and Springer International Publishing Switzerland 2016
Abstract In this paper, we introduce and analyze a fundamental strongly regular equivalence relation on a hypermodule over a hyperring which is the smallest equivalence relation such that the quotient is cyclic module over a (fundamental) ring. Then we state the conditions that is equivalent with the transitivity of this relation. Finally, a characterization of the derived hypermodule (with canonical hypergroup) over a Krasner hyperring has been considered. Résumé Au cours de cet article, nous introduisons et analysons une relation d’équivalence fondamentale fortement réguliére sur un hypermodule sur un hyperanneau qui est la plus petite relation d’équivalence vérifiant la propriété que le quotient est un module cyclique sur un anneau (fondamental). Puis nous énonçons des conditions qui sont équivalentes á la transitivitéé de cette relation. Enfin, nous considérons aussi une caractérisation de l’hypermodule dérivé (avec l’hypergroupe canonique) sur un hyperanneau de Krasner. Keywords Cyclic module · Hypermodule · Hyperring · Strongly regular relation Mathematical Subject Classification 16Y99 · 20N20
B
B. Davvaz [email protected]; [email protected] S. Mirvakili [email protected] P. Ghiasvand [email protected]
1
Department of Mathematics, Payame Noor University, Tehran, Iran
2
Department of Mathematics, Yazd University, Yazd, Iran
123
S. Mirvakili et al.
1 Introduction A hypergroup in the sense of Marty [9] is a nonempty set H endowed by a hyperoperation ∗ : H × H → ℘ ∗ (H ), the set of all nonempty subset of H , which satisfies the associative law and the reproduction axiom. Suppose that (H, ·) and (H , ◦) are two semihypergroups. A function f : H → H is called a homomorphism if f (a · b) ⊆ f (a) ◦ f (b) for all a and b in H . We say that f is a good homomorphism if for all a and b in H , f (a · b) = f (a) ◦ f (b). If (H, ·) is a hypergroup and ρ ⊆ H × H is an equivalence relation, then for all nonempty subsets A, B of H we set A ρ B if and only if aρb, for all a ∈ A, b ∈ B. The relation ρ is called strongly regular on the right (on the left) if x ρ y ⇒ a · x ρ a · y (x ρ y ⇒ x · a ρ y · a, respectively), for all (x, y, a) ∈ H 3 . Moreover, ρ is called strongly regular if it is strongly regular on the right and on the left. Let H be a hypergroup and ρ an equivalence relation on H . A hyperoperation ⊗ is defined on H/ρ by ρ(a) ⊗ ρ(b) = {ρ(x)|x ∈ ρ(a) ◦ ρ(b)}. If ρ is strongly regular, then it readily follows that ρ(a) ⊗ ρ(b) = {ρ(x)|x ∈ a ◦ b}. It is well known for ρ strongly regular that H/ρ, ⊗ is a group, that is, ρ(a) ⊗ ρ(b) = ρ(c) for all c ∈ a ◦ b. Basic definitions and propositions about the hyperstructures are found in [3,5]. Krasner [8] has studied the notion of hyperfield, hyperring, and then some researchers. In [5] there are sev
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