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EMBEDDING OF THE LATTICE OF IDEALS OF A RING INTO ITS LATTICE OF FUZZY IDEALS | ||
| Iranian Journal of Fuzzy Systems | ||
| مقاله 6، دوره 6، شماره 3، دی 2009، صفحه 65-71 اصل مقاله (161.75 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22111/ijfs.2009.201 | ||
| نویسنده | ||
Iffat Jahan 
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| Department of Mathematics, Ramjas College, University Of Delhi, Delhi, India | ||
| چکیده | ||
| 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 $C_{s}^{r} (I(R))$. | ||
| کلیدواژه‌ها | ||
| Algebra؛ Ideal of a ring؛ Morphism؛ Embedding؛ Lattice | ||
| مراجع | ||
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[1] N. Ajmal and K. V. K. Thomas, The lattice of fuzzy subgroups and fuzzy normal subgroups, Information Sci., 76 (1994), 1-11. [2] T. Head, A metatheorem for deriving fuzzy theorems from crisp versions, Fuzzy Sets and Systems, 73 (1995), 349-358. [3] L. Wangjin, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (1982), 133-139. [4] D. S. Malik and J. N. Mordeson, Radicals of fuzzy ideals, Information Sci., 65 (1992) 23, 239-252. [5] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517. [6] A. Weinberger, Embedding lattices of fuzzy subalgebras into lattices of crisp sub-algebras, Information Sci., 108 (1998), 51-70. | ||
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