On Weakly Prime Fuzzy Ideals of Commutative Rings

Abstract

In this paper, we present a new notion of fuzzy ideals : called weakly prime fuzzy ideal. Let R be a commutative ring with non-zero identity. A nonconstant fuzzy ideal µ of R is called weakly prime fuzzy ideal if 0_t != x_r y_s ∈ µ implies x_r ∈ µ or y_s ∈ µ for all t ∈ (0, µ(0)]. We investigate some properties of this notion. Morever, it is established relations between weakly prime ideals and weakly prime fuzzy ideals of commutative rings.

Keywords

• weakly prime fuzzy ideals,prime fuzzy ideals

References

1. [1] Anderson D. D. and Smith E., “Weakly prime ideals”, Houston J. Math. 29(4) (2003) : 831-840.
2. [2] Atani S. E. and Farzalipour F., “On weakly primary ideals”, Georgian Math. J. 12(3) (2005) : 423-429.
3. [3] Dixit V.N., Kumar R. and Ajmal N., “Fuzzy ideals and fuzzy prime ideals of a ring”, Fuzzy Sets and Systems 44 (1991) : 127-138.
4. [4] Ersoy B.A., “A Generalization of Cartesian Product of Fuzzy Subgroups and Ideals”, Journal of Applied Sciences 3 (2003) : 100-102.
5. [5] Liu W.J., “Fuzzy invariant subgroups and fuzzy ideals”, Fuzzy Sets and Systems 8 (1982) : 133-139.
6. [6] Martinez L., “Fuzzy subgroups of fuzzy groups and fuzzy ideals of fuzzy rings”, J. Fuzzy Math. 3 (1995) : 833-849.
7. [7] Mukherjee T.K. and Sen M.K., “Prime fuzzy ideals in rings”, Fuzzy Sets and Systems 32 (1989) : 337-341.
8. [8] Zadeh L.A., “Fuzzy sets”, Inform and Control 8 (1965) : 338-353.