RESIDUAL OF IDEALS OF AN L-RING

Document Type : Research Paper

Author

ATMA RAM SANATAN DHARMA COLLEGE, UNIVERSITY OF DELHI, DHAULA KUAN, NEW DELHI – 110021, INDIA

Abstract

The concept of right (left) quotient (or residual) of an ideal η by an
ideal ν of an L-subring μ of a ring R is introduced. The right (left) quotients are
shown to be ideals of μ . It is proved that the right quotient [η :r ν ] of an ideal
η by an ideal ν of an L-subring μ is the largest ideal of μ such that
[η :r ν ]ν ⊆ η . Most of the results pertaining to the notion of quotients
(or residual) of an ideal of ordinary rings are extended to L-ideal theory of
L-subrings.

Keywords


[1] N. Ajmal and A. S. Prajapati, Prime radical and primary decomposition of ideals in an
L-subring, Communicated.
[2] N. Ajmal and S. Kumar, Lattice of subalgebras in the category of fuzzy groups, The
Journal of Fuzzy Mathematics , 10 (2) (2002), 359-369.
[3] G. Birkhoff, Lattice theory, American Mathematical Soceity, Providence, Rhode Island
1967.
[4] D. M. Burton, A first course in rings and ideals, Addison-Wesley, Reading,
Massachusetts, 1970.
[5] D. S. Malik and J. N. Mordeson, Fuzzy prime ideals of rings, FSS, 37 (1990), 93-98.
[6] D. S. Malik and J. N. Mordeson, Fuzzy maximal, radical, and primary ideals of a ring,
Inform. Sci., 53 (1991), 237-250.
[7] D. S. Malik and J. N. Mordeson, Fuzzy primary representations of fuzzy ideals, Inform.
Sci., 55 (1991), 151-165.
[8] D. S. Malik and J. N. Mordeson, Radicals of fuzzy ideals, Inform. Sci., 65 (1992), 239-
252.
[9] D. S. Malik, J. N. Mordeson and P. S. Nair, Fuzzy normal subgroups in fuzzy subgroups,
J. Korean Math. Soc., 29 (1992), 1-8.
[10] D. S. Malik, and J. N. Mordeson, R-primary representation of L-ideals, Inform, Sci., 88
(1996), 227-246.
[11] J. N. Mordeson, L-subspaces and L-subfield, Centre for Research in Fuzzy Mathematics
and Computer Science, Creighton University, USA. 1996.
[12] J. N. Mordeson and D. S. Malik, Fuzzy commutative algebra, World Scientific Publishing
Co. USA. 1998.
[13] A. S. Prajapati and N. Ajmal, Maximal ideals of L-subring, The Journal of Fuzzy
Mathematics (preprint).
[14] A. S. Prajapati and N. Ajmal, Maximal ideals of L-subring II, The Journal of Fuzzy
Mathematics (preprint).
[15] A. S. Prajapati and N. Ajmal, Prime ideal, Semiprime ideal and Primary ideal of an
L-subring, Communicated.
[16] G. Szasz, Introduction to lattice theory, Academic Press, New York and London, 1963.
[17] Y. Yandong, J. N. Mordeson and S.-C. Cheng, Elements of L-algebra, Lecture notes in
Fuzzy Mathematics and Computer Science 1, Center for Research in Fuzzy Mathematics
and Computer Science, Creighton University, USA. 1994.