On&-primary hiperideals and fuzzy hiperideals expansions

Abstract

In this thesis we study the concepts of hypergroup, hyperring and hyperideals. A commutative ring is a set with two binary operations; addition and multiplication. In a hyperring we have a set with two hyperoperations; hyperaddition and hypermultiplication. In Krasner's Hyperring the addition is a hyperoperation, but the multiplication is a binary operation. In our study we concentrate on a hyperring at the sense of Krasner. We use Krasner's method to construct a hyperring and give examples of prime, primary and maximal hyperideals. Also we discuss a hyperring homomorphism, hyperring isomorphism theorems and Chinese remainder theorem for hyperideals. Then, we study extension and contraction of hyperideals. In the fourth chapter the main result is about the primary hyperideals. We show that the primary decomposition of a hyperideal is not unique. Finally, we present a new concept of δ-primary hyperideals expansion of semihyperrings and δ-primary fuzzy hyperidels expansion of semihyperrings.