Bipolar Fuzzy Ideals and Prime Ideals in Left Almost Rings -by Nita Diarningrum, Noor Hidayat and Abdul Rouf Alghofari

# Bipolar Fuzzy Ideals and Prime Ideals in Left Almost Rings-by Nita Diarningrum, Noor Hidayat and Abdul Rouf Alghofari

Abstract: A bipolar fuzzy subset is an extension of a fuzzy subset, that is an extension from mapping $f:R\rightarrow [0,1]$ to mapping $f:R\rightarrow[-1,1]$. Bipolar fuzzy has been studied in rings, but no research has been conducted on LA-ring. A non empty set $(R,+,∙)$ is called an LA-ring if $(R,+)$ is an LA-group and $(R,.)$ is an LA-semigroup. In this paper, we introduce the definitions and properties of bipolar fuzzy ideals, bipolar fuzzy completely prime ideals, and weakly bipolar fuzzy completely prime ideals in LA-rings. Ideal from bipolar fuzzy subset of the form $\gamma_I, t^{\prime} A_{M},$ and bipolar fuzzy ideal from Cartesian product of bipolar fuzzy ideals are also given. Moreover, in this paper we study relationships between ideal and bipolar fuzzy ideal in LA-rings and the relationship between bipolar fuzzy ideal and bipolar fuzzy completely prime ideal in LA-rings.

Keywords: Bipolar fuzzy, bipolar fuzzy ideal, bipolar fuzzy prime ideal and left almost ring.