Shahrood University of Technology Journal of Algebraic Systems 2345-5128 12 1 2024 09 01 NEW FUNDAMENTAL RELATIONS IN HYPERRINGS AND THE CORRESPONDING QUOTIENT STRUCTURES 21 41 2833 10.22044/jas.2022.10071.1501 EN Peyman Ghiasvand Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran. Saeed Mirvakili Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran. Bijan Davvaz Department of Mathematical Sciences, Yazd University, Yazd, Iran. Journal Article 2020 09 15 ‎‎In this article‎, ‎we introduce and analyze the smallest‎ ‎equivalence binary relation $\chi ^{*}$ on a hyperring $R$ such‎ that the quotient $R/\chi ^{*}$‎, ‎the set of all equivalence‎ ‎classes‎, ‎is a commutative ring with identity and of‎ characteristic $m$‎. ‎‎‎‎The ‎characterizations‎ of‎ ‎commutative rings with identity via strongly regular relations‎ is investigated and some properties on the topic are presented‎. ‎Moreover‎, ‎we introduce a new strongly regular relation‎ $\sigma^{*}_{p}$ such that the quotient structure is a $p$-‎ring.‎ Moreover, we introduce a new strongly regular relation $\sigma^{*}_{p}$ such that the quotient structure is a $p$-ring. https://jas.shahroodut.ac.ir/article_2833_457f4c8b5ef1a32d867659bb3549f982.pdf