A common generalization of fuzzy primes

Abstract

Let $R$ be a commutative ring with identity. Let $FI(R)$ be the setof all fuzzy ideals of $R$ and $\varphi :FI(R)\text{rightarrowFI}(R)\cup \{0_R\}$ be a function. We introduce the concept of fuzzy$\varphi$-prime ideals of $R$ and study some of its properties. It willbe shown that under additional conditions fuzzy $\varphi$-primenessimplies fuzzy primeness. We also prove that in the decomposablerings fuzzy ${\varphi }_{(1)}$-primes and fuzzy primes coincide. Thebehavior of this concept with fuzzy localization and fuzzy quotientis also studied.