EMBEDDING OF THE LATTICE OF IDEALS OF A RING INTO ITS LATTICE OF FUZZY IDEALS

Document Type : Research Paper

Author

Department of Mathematics, Ramjas College, University Of Delhi, Delhi, India

Abstract

We show that the lattice of all ideals of a ring $R$ can be embedded in the lattice of all its fuzzy
ideals in uncountably many ways. For this purpose, we introduce the concept of the generalized
characteristic function $\chi _{s}^{r} (A)$ of a subset $A$ of a ring $R$ for
fixed $r , s\in [0,1] $ and show that $A$ is an ideal of $R$ if, and only if, its generalized
characteristic function $\chi _{s}^{r} (A)$ is a fuzzy ideal of $R$. We also
show that the set of all generalized characteristic functions $C_{s}^{r}
(I(R))$ of the members of $I(R)$ for fixed $r , s\in [0,1] $ is a
complete sublattice of the lattice of all fuzzy ideals of $R$ and establish
that this latter lattice is generated by the union of all
its complete sublattices $C_{s}^{r} (I(R))$.

Keywords

References

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Iranian Journal of Fuzzy Systems
Volume 6, Issue 3 - Serial Number 3
September and October 2009
Pages 65-71

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