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 in the lattice of all its fuzzyideals in uncountably many ways. For this purpose, we introduce the concept of the generalizedcharacteristic function \chi _{s}^{r} (A) of a subset A of a ring R forfixed r , s\in [۰,۱] and show that A is an ideal of R if, and only if, its generalizedcharacteristic function \chi _{s}^{r} (A) is a fuzzy ideal of R. We alsoshow that the set of all generalized characteristic functions C_{s}^{r}(I(R)) of the members of I(R) for fixed r , s\in [۰,۱] is acomplete sublattice of the lattice of all fuzzy ideals of R and establishthat this latter lattice is generated by the union of allits complete sublattices C_{s}^{r} (I(R)).

کلیدواژه ها:

Algebra ، Ideal of a ring ، Morphism ، Embedding ، Lattice