Embedding of the Lattice of Ideals of a Ring into its Lattice of Fuzzy Ideals

We show that the lattice of all ideals of a ring R can be embedded into its lattice of all fuzzy ideals in uncountabily many ways. For this purpose we introduce the concept of a generalized characteristic function of a subset A of a ring R for each fixed pair r, s [0, 1] and show that A is an ideal of R if and only if its generalized characteristic function is a fuzzy ideal of R. Also, we show that the set of all generalized characteristic functions of the members of I (R) for a fixed pair r, s [0, 1] is a complete sublattice of the lattice of all fuzzy ideals of R. We establish that the lattice of all fuzzy ideals of R is generated by the union of all its complete sublattices.