On Weakly S-2 Prime Hyperideals of Multiplicative Hyperrings

Suppose that R is a commutative multiplicative hyperring with identity and \( S\subseteq R \) is a multiplicatively closed subset. In this paper, we introduce and study the notions of weakly 2- prime hyperideals and weakly S-2 prime hyperideals, which extend the concepts of prime, 2- prime , weakly prime, weakly S - prime, and S-2 prime hyperideals in multiplicative hyperrings. A proper I of R is called a weakly 2- prime hyperideal if whenever \( \left\{0\right\}\neq x\circ y\subseteq I \) for \( x,y\in R \), then either \( x^2\subseteq I \) or \( y^2\subseteq I \). Furhermore, a proper hyperideal I with \( I\cap S=\emptyset \) is said to be a weakly S-2 prime hyperideal if there exists an element \( s\in S \) such that for all \( x,y\in R \) with \( \left\{0\right\}\neq x\circ y\subseteq I \), we have \( s\circ x^2\subseteq I \) or \( s\circ y^2\subseteq I \). We examine various basic properties of these hyperideals and investigate their characterizations under good hyperring homomorphisms, quotient hyperrings, and fraction hyperrings \( S^{-1}R \). In addition, we generalize the classical 2- prime avodience theorem to weakly 2-prime and weakly S-2 prime hyperideals, and we delineate the structural relationships among prime, 2- prime, weakly 2- prime, weakly S- prime, S-2 prime, weakly \( S-2 \) prime hyperideals.