A note on distributivity of the lattice of $L$-ideals of a ring

Many studies have investigated the lattice of fuzzy substructures of algebraic structures such as groups and rings. In this study, we prove that the lattice of $L$-ideals of a ring is distributive if and only if  the lattice of its ideals is distributive, for an infinitely $\vee$-distributive lattice $L$.

___

  • N. Ajmal and K.V. Thomas, The lattices of fuzzy subgroups and fuzzy normal subgroups, Inform. Sci. 76, 1–11, 1994.
  • N. Ajmal and K.V. Thomas, The lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 74, 371–379, 1995.
  • N. Ajmal and K.V. Thomas, The lattices of fuzzy normal subgroups is modular, Inform. Sci. 83, 199–218, 1995.
  • D. Bayrak and S. Yamak, The lattice of generalized normal L-subgroups, J. Intell. Fuzzy Syst. 27, 1143–1152, 2014.
  • D. Bayrak and S. Yamak Distributivity and pseudocomplementation of lattices of generalized L-subgroups, Int. J. Algebra Stat. 5, 107–114, 2016.
  • G. Birhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ., Rhode Island, 1967.
  • N. Gao, Q. Li and Z. Li, When do L-fuzzy ideals of a ring generate a distributive lattice?, Open Math. 4 (1), 531–542, 2016.
  • J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18, 145–174, 1967.
  • T. Head, A meta theorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and Systems 73, 349–358, 1995.
  • I. Jahan, Modularity of Ajmal for the lattices of fuzzy ideals of a ring, Iran. J. Fuzzy Syst. 5, 71–78, 2008.
  • I. Jahan,The lattice of L-ideals of a ring is modular, Fuzzy Sets and Systems 199, 121–129, 2012.
  • R. Kumar, Non-distributivity of the lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 97, 393–394, 1998.
  • S. Majumdar and Q.S. Sultana, The lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 81, 271–273, 1996.
  • J.N. Mordeson and D.S. Malik, Fuzzy Commutative Algebra, World Scientific, 1998.
  • M. Tarnauceanu, Distributivity in lattices of fuzzy subgroups, Inform. Sci. 179, 1163– 1168, 2009.
  • L.A. Zadeh, Fuzzy sets, Inform. Control 8, 338–353, 1965.
  • Q. Zhang, The lattice of fuzzy (left, right) ideals of a ring is modular, Fuzzy Sets and Systems 125, 209–214, 2002.
  • Q. Zhang and G. Meng, On the lattice of fuzzy ideals of a ring, Fuzzy Sets and Systems 112, 349–353, 2000.