The notion of Multiplicative hyperrings is an important class of algebraic hyperstructures which generalize rings where the multiplication is a hyperoperation, while the addition is an operation. Let R be a commutative Multiplicative hyperring and α,∈,End(R). A proper hyperideal I of R is called α,-prime if x ◦,y ⊆,I for some x, y ∈,R then x ∈,I or α, (y) ∈,I. Indeed, the α,-prime hyperideals are a new generalization of prime hyperideals. In this paper, we aim to study α,-prime hyperideals and give the basic properties of this new type of hyperideals.

‎Let $G$ be a group with identity $e$ ‎and ‎‎$‎R$ be a Multiplicative ‎hyperring‎‎. ‎W‎e introduce and study the notions of graded $2$-absorbing and graded $2$-absorbing primary hyperideals of a graded Multiplicative hyperring $R$ which are generalizations of prime ‎hyperideals‎‎. ‎We present basic properties and characterizations of these graded hyperideals and homogeneous components‎. Among various results we prove that ‎the intersection of ‎‎‎two ‎graded‎ prime ‎hyperideals ‎is a‎ ‎graded ‎‎$‎2‎‎$‎-absorbing ‎hyperideal.‎

In a Multiplicative hyperring, the multiplication is a hyperoperation, while the addition is a binary operation. In this paper, the notion of derivation on Multiplicative hyperrings is introduced and some related properties are investigated.

Let G be a group with identity e and R be a Multiplicative hyperring‎. ‎W‎e introduce the concept of G-graded Multiplicative hyperring R and present some ‎new‎ results and examples‎. ‎This ‎‎‎‎‎article ‎aim ‎is ‎‎‎to ‎introduce ‎and ‎study‎ ‎‎‎‎‎‎‎‎‎graded ‎‎prime ‎and ‎‎graded ‎‎p‎rimary ‎hy‎perideals ‎which ‎are‎ ‎different ‎generalizations ‎of‎ ‎‎prime ‎and‎‎ ‎‎‎‎p‎rimary ‎hyperideals. ‎Several‎ ‎basic ‎properties‎, ‎examples and characterizations of graded ‎prime (graded ‎primary)‎ hyperideals of a graded Multiplicative hyperring R are ‎presented ‎such ‎as ‎investigating‎ of this structure under homogeneous components‎, ‎graded hyperring homomorphisms‎, ‎quotient graded hyperrings and fundamental relations‎.

Our main purpose of this paper is to study the relationship between Multiplicative maps and almost Multiplicative maps between probabilistic normed algebras. We first derive some properties of invertible elements and their relation with Multiplicative maps. Then we show that every complex homomorphism on elements whose probabilistic norm is equal to 1, is bounded. In the following, we give an open problem about the functionally continuous of unital commutative probabilistic Banach algebra. Finally, we prove that every almost Multiplicative map that is not a Multiplicative map is continuous.

‎Krasner $F^{(m,n)}$-hyperrings were introduced and  investigated by Farshi and Davvaz. In this paper, our purpose is to  define and characterize  three particular classes of $F$-hyperideals in a Krasner $F^{(m,n)}$-hyperring, namely prime $F$-hyperideals, maximal $F$-hyperideals and  primary $F$-hyperideals, which extend similar concepts of ring context. Furthermore, we  examine the relations between these structures. Then a number of major conclusions are given to explain the general framework of these structures.