fuzzy commutative algebra || back matter
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Bibliography
1. Abadi, H. H and Zahedi, M. M., A density theorem on fuzzy prime spectrum of a ring, preprint.
2. Abadi, H. H and Zahedi, M. M., Some results on fuzzy prime spectrum of a ring, FSS 77 (1996) 235 - 240.
3. Abdukhalikov, K. S., The dual of a fuzzy subspace, FSS 82 (1996) 375 -381.
4. Abdukhalikov, K. S., Tulenbaev, M. S., and Umirbaev, U. U., On fuzzy bases of vector spaces, FSS 63 (1994) 201 - 206.
5. Abou-Zaid, Salah, On fuzzy ideals and fuzzy quotient rings of a ring, FSS 59 (1993) 205 - 210.
6. Ahsan, J., Khan, M. F., Shabir, M. and Zaman, N., Rings characterized by their fuzzy submodules, Inform. Sci. 74 (1993) 247-264.
7. Ajmal, N., Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups, FSS 61 (1994) 329 - 339.
8. Ajmal, N. and Thomas, K. V., The lattice of fuzzy subgroups and fuzzy normal subgroups, Inform. Sci. 76 (1994) 1 -11 .
9. Ajmal, N., The lattice of fuzzy normal subgroups is modular, Inform Sci. 83 (1995) 199 - 209.
10. Ajmal, N., Fuzzy subgroups with sup property, Inform Sci. 93 (1996) 247 - 264.
11. Ajmal, N. and Thomas, K. V., The lattice of fuzzy ideals of a ring, FSS 74 (1995) 371-379.
12. Ajmal, N. and Thomas, K. V., A complete study of the lattices of fuzzy congruences and fuzzy normal subgroups, Inform. Sci., 82 (1995) 198 -218.
Fuz
zy C
omm
utat
ive
Alg
ebra
Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
385
13. Akgul, M., Some properties of fuzzy groups, J. Math. Anal. Appl, 133 (1988) 93-100.
14. Alkhamees, Y., Fuzzy cyclic subgroups and fuzzy cyclic p-groups, J. Fuzzy Math. 3 (1995) 911 - 919.
15. Alkhamees, Y. and Mordeson, J. N., Fuzzy principal ideals and fuzzy simple field extensions, FSS, 96 (1998) 247-253.
16. Alkhamees, Y. and Mordeson, J. N., Fuzzy localized subrings, Inform. Sci, 99 (1997) 183-193.
17. Alkhamees, Y. and Mordeson, J. N., Local examination of fuzzy intersection equations, FSS 98 (1998) 249-254.
18. Alkhamees Y. and Mordeson, J. N., Reduced fields, primitive fields and fuzzy Galois theory, preprint.
19. Asaad, M., Groups and fuzzy subgroups, FSS 39 (1991) 323-328.
20. Asaad M. and Abou-Zaid S., Fuzzy subgroups of nilpotent groups, FSS 60 (1993) 321 - 323.
21. Assad, M. and Abou-Zaid, S., A contribution to the theory of fuzzy subgroups, FSS (1996) 355 - 369.
22. Balbes, R. and Dwinger, P., Distributive Lattices, University of Missouri Press, 1974.
23. Barnes, W. E., Introduction to Abstract Algebra, D. C. Heath, Boston, 1963.
24. Bhakat, S. K. and Das, P., Fuzzy subrings and ideals redefined, FSS 81 (1996) 383-393.
25. Bhambri S. K., Kumar, R. and Kumar, P., On fuzzy primary submodules, Bull. Cal. Math. Soc. 86 (1994) 445-452.
26. Bhambri S. K., Kumar, R. and Kumar, P., Fuzzy prime submodules and radical of a fuzzy submodule, Bull. Cal. Math. Soc. 87 (1995) 163-168.
27. Birkhoff, G., Lattice Theory, American Mathematical Society, Providence, Rhode Island (1967).
28. Biswas, R., Fuzzy fields and fuzzy linear spaces redefined, FSS 33 (1989) 257 - 259.
Fuz
zy C
omm
utat
ive
Alg
ebra
Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
386
29. Borzoee, R. A. and Zahedi, M. M., Fuzzy algebraic extension generated by a fuzzy subset and maximal fuzzy algebraic extension, J. Fuzzy Math. I (1993) 649 - 657.
30. Bourbaki, N. (transl. by P. M. Cohn and J. Howie), Elements of Mathematics, Algebra II, Chapters 4 - 7 , Springer - Verlag, Berlin, New York (1981).
31. Bourbaki, N., Elements of Mathematics, Commutative Algebra, Springer-Verlag, New York, 1989, Chaps. 1-7.
32. Chen De-Gang and Gu Wen-Xiang, Product structure of fuzzy factor groups, FSS 60 (1993) 2229 - 232.
33. Cox, D., Little, J., and O'Shea, D., Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer-Verlag, 1992.
34. Das, P., Fuzzy vector spaces under triangular norms, FSS 25 (1988) 73 -85 .
35. Das, P. S., Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264 - 269.
36. Deveney, J. K., An intermediate theory for a purely inseparable Galois theory, Trans. Amer. Math. Soc. 198 (1974) 287 - 295.
37. Deveney, J. K. and Mordeson, J. N., Splitting and modularly perfect fields, Pacific J. Math. 83 (1979) 45-54.
38. Deveney, J. K. and Mordeson, J. N., Distinguished subfields of intermediate fields, Canad. J. Math. 33 (1981) 1085-1096.
39. Deveney, J. K. and Mordeson, J. N., Uniqueness of subfields, Canad. Bull. Math. 29 (1986) 191-196.
40. Deveney, J. K. and Mordeson, J. N., Inseparable extensions and primary abelian groups, Arch. Math. 33 (1979) 538 - 545.
41. Dib, K. A., On fuzzy subspaces and fuzzy group theory, Inform. Sci. 80 (1994) 253 - 282.
42. Dib, K. A., Galhum, N. and Hassam A. A. M., Fuzzy rings and fuzzy ideals, J. Fuzzy Math., 4 (1996) 245-261.
43. Divinsky, N. J., Rings and Radicals, University of Toronto Press (1965).
Fuz
zy C
omm
utat
ive
Alg
ebra
Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
387
44. Dixit, V. N., Kumar, R., and Ajmal, N., Fuzzy ideals and fuzzy prime ideals of a ring, FSS 44 (1991) 127 - 138.
45. Dixit, V. N., Kumar, R., and Ajmal, N., On fuzzy rings, FSS 49 (1992) 205 - 213.
46. Eke, A. C. and Eke, B. I., Special fuzzy generators and intermediate fuzzy field extensions, FSS 64 (1994) 407 - 413.
47. Eslami, E. and Mordeson., J. N., Structure of fuzzy subrings, Inform. Sci. 76 (1994) 57 - 65.
48. Eslami, E. and Mordeson., J. N., Completions and fuzzy power series subrings, FSS 82 (1996) 97 - 102.
49. Fuchs, L., Infinite Abelian Groups I, Pure and Applied Math., Vol. 36, Academic Press New York 1070.
50. Garmendia, A., Molero, M. and Salvador, A., T-fuzzy prime ideals, FSS 80 (1996) 245-254.
51. Garzon, M. and Muganda, G. C., Free fuzzy groups and fuzzy group presentations, FSS 48 (1992) 249 - 255.
52. Goguen, J. A., L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145 - 174.
53. Golan, J. S., Norm, semirings, and power algebra, in: Ramanujan Conference on Algebra and Its Applications, Madra, India (to appear).
54. Golan, J. S., Making modules fuzzy, FSS 32 (1989) 91-94.
55. Gu Wenxiang and Lu Tu, Fuzzy linear spaces, FSS 49 (1992) 377 - 380.
56. Gu Wenxiang and Lu Tu, Fuzzy algebras over fuzzy fields redefined, FSS 53 (1993) 105 - 107.
57. Gu Wenxiang and Lu Tu, Some properties of divisible and pure fuzzy subgroups, Fuzzy Syst & Math. 6 (1) (1992) 48 - 53.
58. Gu Wenxiang and Lu Tu, The properties of fuzzy divisible groups, FSS 56 (1993) 195 - 198.
59. Gupta, K. C. and Kantroo, M. K., The intrinsic product of fuzzy subsets of a ring, FSS 57 (1993) 103-110.
60. Gupta, K. C. and Kantroo, M. K., The nil radical of a fuzzy ideal, FSS 59 (1993) 87 - 93.
Fuz
zy C
omm
utat
ive
Alg
ebra
Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
388
61. Head, T., A metatheorem for deriving fuzzy theorems from crisp versions, FSS 73 (1995) 349-358.
62. Head, T., Erratum to "A metatheorem for deriving fuzzy theorems from crisp versions," FSS 79 (1996) 277-278.
63. Head, T., Embedding lattices of fuzzy subgroups into lattices of crisp subgroups, Proceedings, Biennial Conference of the North American Fuzzy Information Processing Society - NAFIPS 1996.
64. Heerema, N and Tucker, D., Modular field extensions, Proc. Amer. Math. Soc. 53 (1975) 301 - 306.
65. Jaballah, A. and Mordeson, J. N., Minimal generating systems for fuzzy ideals, Soochow J. Math., 21 (1995) 183 - 192.
66. Jacobson, N., Lectures in Abstract Algebra, Vol. II, Linear Algebra, D. Van Nostrand Company, Inc. Princeton, New Jersey (1953).
67. Jacobson, N., Lectures in Abstract Algebra, Vol. Ill, Theory of Fields and Galois Theory, D. Van Nostrand Company, Inc. Princeton, New Jersey (1964).
68. Jacobson, N., Basic Algebra I, W. H. Freeman and Company, San Francisco (1974).
69. Jacobson, N., Basic Algebra II, W. H. Freeman and Company, San Francisco (1980).
70. Jiang Huabiao, The direct sum of fuzzy rings, J. National Defense Sci. & Technol. Univ. No. 3 (1985) 105 - 110.
71. Jiang Huabiao, Fuzzy ideals and quotient rings, Sci. Exploration, No. 1 (1986) 17 - 24.
72. Jiang Huabiao, Fuzzy subfields on a finite extension field, J. National Defense Sci. & Technol. Univ., No. 1 (1988) 65 - 69.
73. Jianli, Z., Kaiquan, S., and Mingshan, Y., Fuzzy modules over fuzzy rings, J. Fuzzy Math. 1 (1993) 531-539.
74. Karpilovsky, G., Topics in Field Theory, North-Holland Mathematics Studies 155, Elsevier Science Publishers B. V., 1989.
75. Katsaras, A. K. and Liu, D. B., Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58 (1977) 135 - 146.
Fuz
zy C
omm
utat
ive
Alg
ebra
Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
389
76. Kim, Jae-Gyeom, Orders of the fuzzy subgroups and fuzzy p-groups, FSS 61 (1994) 225 - 230.
77. Kim, Jae-Gyeom, Fuzzy subgroups having property (*), Inform. Sci. 80 (1994) 235 - 241.
78. Kim, Jae-Gyeom and Kim Han-Doo, A characterization of fuzzy subgroups of some abelian groups, Inform. Sci. 80 (1994) 243 - 252.
79. Kim Jae-Gyeom, Fuzzy orders relative to fuzzy subgroups, Inform. Sci. 80 (1994) 341 - 348.
80. Kim, Jae-Gyeom, Fuzzy subgroups and minimal fuzzy p-subgroups, J. Fuzzy Math., 2 (1994) 913 - 921.
81. Kim, Jae-Gyeom, Commutative fuzzy sets and nilpotent fuzzy groups, Inform. Sci. 83 (1995) 161 - 174.
82. Kim, Jae-Gyeom, Lattices of fuzzy subgroupoids, fuzzy submonoids, and fuzzy subgroups, Inform. Sci., 91 (1996) 77 - 93.
83. Kim, Jae-Gyeom, Join-reducibility of fuzzy subgroups, Inform. Sci., 91 (1996) 211 - 225.
84. Kohli, J. K. and Kumar, R., Fuzzy prime spectrum of a ring II, FSS 59 (1993) 223 - 230.
85. Kreimer, H. and Heerema, N., Modularity vs. separability for field extensions, Canad. J. Math. 27 (1975) 1176 - 1182.
86. Kumar, R., Fuzzy semiprimary ideals of rings, FSS 42 (1991) 263 - 272.
87. Kumar, R., Fuzzy irreducibile ideals in rings, FSS 42 (1991) 360 - 379.
88. Kumar, R., Fuzzy nil radicals and fuzzy primary ideals, FSS 43 (1991) 81 - 91.
89. Kumar, R., Fuzzy vector spaces and fuzzy cosets, FSS 45 (1992) 109 -116.
90. Kumar, R., Fuzzy prime spectrum of a ring, FSS 46 (1992) 147 - 154.
91. Kumar, R., Certain fuzzy ideals of rings redefined, FSS 46 (1992) 251 -260.
92. Kumar, R., Fuzzy cosets and some fuzzy radicals, FSS 46 (1992) 261 -265.
Fuz
zy C
omm
utat
ive
Alg
ebra
Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
390
93. Kumar, R., Fuzzy subgroups, fuzzy ideals, and fuzzy cosets: Some properties, FSS 48 (1992) 267 - 274.
94. Kumar, R., On fuzzy irreducible ideals and homomorphisms (Short Communication), FSS 50 (1992) 355 - 356.
95. Kumar, R., Fuzzy nil radicals and fuzzy semiprimary ideals, FSS 51 (1992) 345 - 350.
96. Kumar, R., Fuzzy ideals and fuzzy semiprime ideals, Inform. Sci. 66 (1992) 43 - 52.
97. Kumar, R., On the fuzzy radical of a fuzzy ideal (Short Communication), FSS 53 (1993) 233 - 235.
98. Kumar, R., On the dimension of a fuzzy subspace, FSS 54 (1993) 229 -234.
99. Kumar, R. Bhambri, S. K. and Kumar P., Fuzzy submodules: some analogues and deviations, FSS 70 (1995) 125-130.
100. Kumbhojkar, H. V. and Bapat, M. S., Not-so-fuzzy fuzzy ideals, FSS 37 (1990) 237 - 243.
101. Kumbhojkar, H. V. and Bapat, M. S., Correspondence theorems for fuzzy ideals, FSS 41 (1991) 213 - 219.
102. Kumbhojkar, H. V. and Bapat, M. S., On semiprime fuzzy ideals, FSS 60 (1993) 219 - 223.
103. Kumbhojkar, H. V. and Bapat, M. S., On prime and primary fuzzy ideals and their radicals, FSS 53(1993) 203-216.
104. Kumbhojkar, H. V., Spectrum of prime fuzzy ideals, FSS 62 (1994) 101 -109.
105. Kumbhojkar, H. V., Some comments on spectrums of prime fuzzy ideals of a ring, FSS 85(1997) 109-114.
106. Kunz, E., Introduction to Commutative Algebra and Algebraic Geometry, Birkhouser Boston, 1985.
107. Kuraoka, T. and Kuroki, N., On fuzzy quotient rings induced by fuzzy ideals, FSS 47 (1992) 381 - 386.
108. Kuraoka, T. and Suzuki, Nobu-Yuki, A simple characterization of fuzzy subgroups, Inform. Sci. 73 (1993) 41 - 55.
Fuz
zy C
omm
utat
ive
Alg
ebra
Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
391
109. Kuroki N., On fuzzy semigroups, Inform. Sci. 53(1991) 203-236.
110. Kuroki N., On fuzzy ideals and fuzzy biideals in semigroups, FSS 5 (1981) 203-215.
111. Lajos S., A note on semilattices of groups, Ada. Sci. Math. (Szeged) 33(1972) 315-317.
112. Lee, K. Y. and Mordeson, J. N., Factorization of fuzzy ideals in Dedekind domains, J. Fuzzy Math. 5(1997) 741-745.
113. Lee, K. Y. and Mordeson, J. N., Fractionary fuzzy ideals and fuzzy in-vertible fractionary fuzzy ideals, J. Fuzzy Math. 5(1997) 875-883.
114. Lee, K. Y. and Mordeson, J. N., Fractionary fuzzy ideals and Dedekind domains, preprint.
115. Li, S. Y., Almost trivial fuzzy vector spaces, J. Fuzzy Math., 3 (1995) 633-636.
116. Liu Wangjin, Fuzzy invariant subgroups and fuzzy ideals, FSS 8 (1982) 133 - 139.
117. Liu Wangjin, Operations on fuzzy ideals, FSS 11 (1983) 31 - 41.
118. Liu Wangjin, L-fuzzy generated ideals and co-prime ideals, J. Sichuan Teachers College (Nat. Sci. Ed.), No. 3 (1985) 37 - 42.
119. Liu Wangjin, L-fuzzy prime ideals and radical ideals, J. Sichuan Teachers College (Nat. Sci. Ed.), No. 3, (1985) 67 - 72.
120. Liu Wangjin, Fuzzy prime ideals and fuzzy radical ideals, Inform. Sci. 46 (1990) 1- 12.
121. Liu Wangjin, On some systems of simultaneous equations in a completely distributive lattice, Inform. Sci. 50 (1990) 185 - 196.
122. Liu Xilong and Yu Yandong, The isomorphisms of fuzzy linear spaces (1), J. Xuzhou Teachers College (Nat. Sci. Ed.), No. 1 (1986) 10 - 15.
123. Liu Xilong and Yu Yandong, The isomorphisms of fuzzy linear spaces (2), J. Xuzhou Teachers College (Nat Sci. Ed.), No. 2 (1986) 14 - 19.
124. Lowen, R., Convex fuzzy sets, FSS 3 (1980) 291 - 310.
125. Lu Tu and Gu Wenxiang, Abelian fuzzy group and its properties, FSS 64 (1994) 415 - 420.
Fuz
zy C
omm
utat
ive
Alg
ebra
Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
392
126. Lu Tu and Gu Wenxiang, Some properties of fuzzy vector spaces, J. Fuzzy Math. 3 (1995) 921-931.
127. Lubczonok, P., Fuzzy vector spaces, FSS 38 (1990) 329 - 343.
128. Ma Jiliang and Yu Chunhai, Fuzzy rings (I), J. Northeast Normal Univ. (Nat. Sci. Ed.), No. 1 (1982) 23 - 38.
129. Majumdar, S. and Sultana Q., The lattice of fuzzy ideals of a ring, FSS 81 (1996) 271-273.
130. Makamba, B. B., Direct products and isomophism of fuzzy subgroups, Inform. Sci. 65 (1992) 33 - 43.
131. Makamba, B. B., Studies in fuzzy groups, Ph. D. thesis, Rhodes Univ., 1992.
132. Makamba, B. B. and Murali, V., Normality and congruence in fuzzy subgroups, Inform. Sci. 59 (1992) 121 - 129.
133. Malik, D. S., Fuzzy ideals of Artinian rings, FSS 37 (1990) 111 - 115.
134. Malik, D. S. and Mordeson, J. N., Fuzzy prime ideals of rings, FSS 37 (1990) 93 - 98.
135. Malik, D.S. and Mordeson, J. N., Fuzzy subfields, FSS 37 (1990) 383 -388.
136. Malik, D. S. and Mordeson, J. N., Fuzzy maximal, radical, and primary ideals of a ring, Inform. Sci. 53 (1991) 237 - 250.
137. Malik, D. S. and Mordeson, J. N., Fuzzy primary representations of fuzzy ideals, Inform. Sci. 55 (1991) 151 - 165.
138. Malik, D. S. and Mordeson, J. N., Fuzzy vector spaces, Inform. Sci. 55 (1991) 271 - 281.
139. Malik, D. S. and Mordeson, J. N., Fuzzy subgroups of abelian groups, Chinese J. Math. (Taipei) 19 (1992) 129 - 145.
140. Malik, D. S. and Mordeson, J. N., Fuzzy direct sums of fuzzy rings, FSS 45 (1992) 83 - 91.
141. Malik, D. S. and Mordeson, J. N., Extensions of fuzzy subrings and fuzzy ideals, FSS 45 (1992) 245 - 251,
142. Malik, D. S. and Mordeson, J. N., Fuzzy homomorphisms of rings, FSS 46 (1992) 139 - 146.
Fuz
zy C
omm
utat
ive
Alg
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Dow
nloa
ded
from
ww
w.w
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scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
393
143. Malik, D. S. and Mordeson, J. N., Radicals of fuzzy ideals, Inform. Sci. 65 (1992) 239 - 252.
144. Malik, D. S. and Mordeson, J. N., 7£-primary representations of L-ideals, Inform, Sci. 88 (1996) 227 - 246.
145. Malik, D. S. and Mordeson, J. N., L-prime spectrum of a ring, NAFIPS 97(1997) 273-278.
146. Malik, D. S., Mordeson, J. N., and Nair, P. S., Fuzzy normal subgroups in fuzzy subgroups, J. Korean Math. Soc. 29 (1992) 1 -8 .
147. Malik, D. S., Mordeson, J. N., and Nair, P. S., Fuzzy generators and fuzzy direct sums of abelian groups, FSS 50 (1992) 193 - 195.
148. Martinez, L., Fuzzy subgroups of fuzzy groups and fuzzy ideals of fuzzy rings, J. Fuzzy Math., 3 (1995) 833-849.
149. Martinez, L., Fuzzy modules over fuzzy rings in connection with fuzzy ideals of fuzzy rings, J. Fuzzy Math., 4 (1996) 843-857.
150. Mashinchi, M. and Mukaidono, M., Generalized fuzzy quotient subgroups, FSS 74 (1995) 245 - 257.
151. Mashinchi, M. and Zahedi, M. M., On L-fuzzy primary submodules, FSS 49 (1992) 231-236.
152. May, W., Commutative Group Algebras, Trans. Amer. Math. Soc. 136 (1969) 139 - 149.
153. McCarthy, P. J., Algebraic Extension of Fields, Blaisdell Publishing Company (1966).
154. McCoy, N. H., Rings and Ideals, The Carus Mathematical Monograms, Number 8, The Mathematical Association of America, 1956.
155. Mordeson, J. N., Remark on coefficient fields in complete local rings, J. Math. Kyoto Univ. 4 (1965) 637 - 639.
156. Mordeson, J. N., Splitting of field extensions, Archiv Math. 26 (1975) 606-610.
157. Mordeson, J. N., Modular extensions and abelian groups, Arch. Math. 36 (1981) 13 - 20.
158. Mordeson, J. N., Fuzzy algebraic field extensions, FSS 45 (1992) 359 -365.
Fuz
zy C
omm
utat
ive
Alg
ebra
Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
QU
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N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
394
159. Mordeson, J. N., Fuzzy-field extensions, FSS 47 (1992) 253 - 264.
160. Mordeson, J. N., Fuzzy transcendental field extensions, Chinese J. Math. (Taipei) 20 (1992) 55 - 68.
161. Mordeson, J. N., Fuzzy subfields of finite fields, FSS 52 (1992) 93 - 96.
162. Mordeson, J. N., Distinguished subfields and splitting of fuzzy subfields, Bull. Cal. Math. Soc. 84 (1992) 121 - 126.
163. Mordeson, J. N., Bases of fuzzy vector spaces, Inform. Sci. 67 (1993) 87 -92 .
164. Mordeson, J. N., Generating properties of fuzzy algebraic structures, FSS 55 (1993) 107 - 120.
165. Mordeson, J. N., Splitting of fuzzy field extensions, J. Nigerian Math. Soc. 11 (1992) 33 - 44.
166. Mordeson, J. N., Fuzzy algebraic varieties, Rocky Mountain J. Math. 23 (1993) 1361 - 1377.
167. Mordeson, J. N., Fuzzy algebraic varieties II, Advances in Fuzzy Theory and Technology Vol. I (1993) (Edited by Paul Wang) 9 - 21.
168. Mordeson, J. N., Fuzzy Galois theory, J. of Fuzzy Math. 1 (1993) 659 -671.
169. Mordeson, J. N., Fuzzy coefficient fields of fuzzy subrings, FSS 58 (1993) 227 - 327.
170. Mordeson, J. N., Bases of fuzzy algebraic structures, FSS 62 (1994) 185 - 191.
171. Mordeson, J. N., Invariants of fuzzy subgroups, FSS 63 (1994) 81 - 85.
172. Mordeson, J. N., Infinite fuzzy Galois theory, Advances in Fuzzy Theory and Technology Vol. II (1994) (Edited by Paul Wang) 99 - 109.
173. Mordeson, J. N., Fuzzy group subalgebras, J. Fuzzy Math. 3 (1995) 69 -81.
174. Mordeson, J. N., Relative p-bases of fuzzy field extensions, Chinese J. Math., 22 (1994) 297 - 308.
175. Mordeson, J. N., Fuzzy intersection equations and primary representations, FSS, 83 (1996) 93 - 98.
Fuz
zy C
omm
utat
ive
Alg
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Dow
nloa
ded
from
ww
w.w
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scie
ntif
ic.c
omby
QU
EE
N M
AR
Y U
NIV
ER
SIT
Y O
F L
ON
DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
395
176. Mordeson, J. N., Commentary: Definition of prime and primary fuzzy ideals, J. Fuzzy Math. 3 (1995) 345 - 349.
177. Mordeson, J. N., Fuzzy group subalgebras II, J. Fuzzy Math.. 3 (1995) 885 - 897.
178. Mordeson, J. N. and Peng, Chang-Shyh, Fuzzy intersection equations, FSS 60 (1993) 77 - 81.
179. Mordeson, J. N. and Peng, Chang-Shyh, Operations on fuzzy graphs, Inform. Sci. (1994) 159 - 170.
180. Mordeson, J. N. and Sen, M. K., Basic fuzzy subgroups, Inform. Sci., 82 (1995) 167 - 179.
181. Mordeson, J. N. and Vinograde B., Extension of certain subfields to coefficient fields in commutative algebras, J. Math. Soc. Japan 17 (1965) 4 7 - 5 1
182. Mordeson, J. N. and Vinograde B., Generators and tensor factors of arbitrary purely inseparable fields, Math. Zeit., 107 (1968) 326-334.
183. Mordeson, J. N. and Vinograde, B., Arbitrary Purely Inseparable Extension Fields, Lecture Notes in Math. 173, Springer-Verlag, Berlin Heidelberg, New York 1970.
184. Morsi, N. N. and Yehia, S. E., Fuzzy-quotient groups, Inform. Sci, 81 (1994) 177 - 191.
185. Morsi, N. N., Note on normal fuzzy subgroups and fuzzy normal series of finite groups, FSS 72 (1995) 379-383.
186. Muganda, G. C , Fuzzy linear and affine spaces, FSS 38 (1990) 365 - 373.
187. Muganda, G. C , Free fuzzy modules and their bases, Inform. Sci.,72 (1993) 65 - 82.
188. Muganda, G. C , On the existence of bases for fuzzy vector spaces, preprint.
189. Muganda, G. C , Fuzzy algebras based on algebraic properties, preprint.
190. Muganda, G. C. and Garzon, M., On the structure of fuzzy groups, Advances in Fuzzy Theory and Technology Vol. I (1993) (Edited by Paul Wang) 23 - 42.
191. Mukherjee, N. P. and Bhattachaxya, P., Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225 - 239.
Fuz
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DO
N o
n 07
/08/
14. F
or p
erso
nal u
se o
nly.
396
192. Mukherjee, T. K. and Sen, M. K., On fuzzy ideals of a ring I, FSS 21 (1987) 99 - 104.
193. Mukherjee, T. K., and Sen, M. K., Prime fuzzy ideals in rings, FSS 32 (1989) 337 - 341.
194. Mukherjee, T. K. and Sen, M. K., Rings with chain conditions, FSS 39 (1991) 117 - 123.
195. Mukherjee, T. K., Sen, M. K. and Roy D., On fuzzy submodules of their radicals, J. Fuzzy Math., 4 (1996) 549-558.
196. Murthy, N. V. E. S., How fuzzy are fuzzy ideals?, J. Fuzzy Math., 4 (1996) 15 - 23.
197. Nagata, M., Note on coefficient fields in complete local rings, Mem. Coll. Sci. Uni. Kyoto 32 (1959 - 60) 91 - 92.
198. Nagata, M., Local Rings, Interscience Publishers #13, John Wiley and Sons Inc., 1962.
199. Nanda, S., Fuzzy fields and linear spaces, FSS 19 (1986) 89 - 94.
200. Nanda, S. Fuzzy algebras over fuzzy fields, FSS 37 (1990) 99 - 103.
201. Nanda, S., Fuzzy linear spaces over valued fields, FSS 42 (1991) 351 -354.
202. Negoita, C. V. and Ralescu, D. A., Applications of Fuzzy Sets to Systems Analysis, Wiley, New York (1975) 54 - 59.
203. Ovchinnikov S., Continuous fuzzy groups, FSS 76 (1995) 253 - 257.
204. Pan, F., Fuzzy finitely generated modules, FSS 21 (1987) 105-113.
205. Pan, F., Exact sequences of fuzzy linear maps, FSS 27 (1988) 317-325.
206. Pan, F., Fuzzy quotient modules, FSS 28 (1988) 85-90.
207. Pan, F., The various structures of fuzzy quotient modules, FSS 50 (1992) 187-192.
208. Peng, B.,The isomorphism of fuzzy groups and some problems of calculation, J. South China Normal Univ. (Nat. Sci. Ed. Math. Phys. Spec. Issue) (1990) 158-162.
209. Passman, D. S., The Algebraic Structure of Group Rings, A. Wiley Interscience Publication, John Wiley Sz Sons, Inc. 1977.
Fuz
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ive
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DO
N o
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/08/
14. F
or p
erso
nal u
se o
nly.
397
210. Petrich, M., Introduction to Semigroups, Charles E. Merrill Publishing Co., A Bell and Howell Co., Ohio, 1973.
211. Ray, S., Isomorphic fuzzy groups, FSS 50 (1992) 201 - 207.
212. Ray, S., The free product of fuzzy subgroups, FSS 50 (1992) 225-235.
213. Ray, S., Analysis of the level subgroups of a fuzzy group, FSS 51 (1992) 323-331.
214. Ray, S., Solvable fuzzy groups, Inform. Sci. 75 (1993) 47 - 61.
215. Ren Yongcai, Fuzzy ideals and quotient rings, Fuzzy Math., 5 (4) (1985) 19 - 26.
216. Rosenfeld, A., Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512 - 517.
217. Rygg, P. T., On mimimal sets of generators of purely inseparable field extensions, Proc. Amer. Math. Soc. 14 (1963) 742 - 745.
218. Salili, S. and Mashinchi, M., On fuzzy isomorphism, J. Fuzzy Math. 4 (1996) 39 - 49.
219. Sanchez, E., Resolution of composite fuzzy relation equations, Inform. and Control 30 (1976) 38-48.
220. Sebastian, S. and Sundar, S. B., Commutative L-fuzzy subgroups, FSS 68 (1994) 115 - 121.
221. Sessa S., On fuzzy subgroups and fuzzy ideals under triangular norms (Short communication), FSS 13 (1984) 95-100.
222. Sidky, F. I., Essential and superfuluous fuzzy submodules, J. Inst. Math. Comput. Sci. Math. Ser. 4 (1991) 103-106.
223. Sidky, F. I. and Khatab, S. A., Nil radical of fuzzy ideal, FSS 47 (1992) 117- 120.
224. Sidky, F. I. and Mishref, M. A. A., Divisible and pure fuzzy subgroups, FSS 34 (1990) 377 - 382.
225. Sidky, F. I. and Mishref, M. A. A., Fuzzy cosets and cyclic and abelian fuzzy subgroups, FSS 43 (1991) 243 - 250.
226. Sidky, F. L. and Mishref, M. A. A., Nil radical of fuzzy ideal, FSS 47 (1992) 117 - 120.
Fuz
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DO
N o
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/08/
14. F
or p
erso
nal u
se o
nly.
398
227. Suzuki, Y., On the construction of free fuzzy groups, J. Fuzzy Math., 2 (1994) 1 - 15.
228. Swamy, U. M. and Raju, D. V., Algebraic fuzzy systems, FSS 41 (1991) 187 - 194.
229. Swamy, U. M. and Raju, D. V., Irreduciblity in algebraic fuzzy systems, FSS 41 (1991) 233 - 241.
230. Swamy, U. M. and Swamy, K. L. N., Fuzzy prime ideals of rings, J. Math. Anal. Appl. 134 (1988) 94 - 103.
231. Sweedier, Structure of inseparable extensions, Annal. of Math. 87(1968) 401 - 410; correction, ibid (2) 89 (1969), 206 - 207.
232. Szasz, G. Introduction to Lattice Theory, Academic Press, New York and London, 1963.
233. Wang, G., Pointwise topology on a complete distributive lattices (I) (in Chinese), J. Shanxi Normal Univ., 1 (1985) 1-17.
234. Wang, Z. D., TL-submodules and TL-linear subspaces, FSS, 68 (1994) 211-225.
235. Wang, Z. D., Primary TL-submodules and P-primary TL-submodules, FSS 88(1997) 237-254.
236. Wang, Z. D. and Yu, Y. -D., TL-subrings and TL-ideals, Part 2: generated TL-ideals, FSS 87 (1997) 209-217.
237. Wang, Z. D. and Yu, Y. -D., TL-subrings and TL-ideals, Part 5: Prime TL-ideals and semiprime TL-ideals, FSS, submitted.
238. Waterhouse, W., The structure of inseparable field extesions, Trans. Amer. Math. Soc. 211 (1975), 39 - 56.
239. Weinberger, A., Embedding lattices of fuzzy subalgebras into lattices of crisp subalgebras, Inform. Sci, 108 (1998) 51-70.
240. Weng Wenhui, Fuzzy subspaces of a linear space, J. Xiamen Univ. (Nat. Sci. Ed.) 21 (1982) 371 - 376.
241. Weng Wenhui, The countable decomposition theorem of fuzzy subspaces, J. Xiamen Univ. (Nat. Sci. Ed.) 27 (1985) 20 - 25.
242. Weng Wenhui, The basis of fuzzy linear space, J. Xiamen Univ. (Nat. Sci. Ed.) 27 (1988) 385 - 386.
Fuz
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DO
N o
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/08/
14. F
or p
erso
nal u
se o
nly.
399
243. Weiss, M.D., Fixed points separation, and induced topologies for fuzzy sets, J. Math. Appl. 50(1975) 142-150.
244. Winter, D. J., The Structure of Fields, Graduate Texts in Mathematics, Springer-Verlag New York Heidelberg Berlin, 1970.
245. Wu, Wangming, Normal fuzzy subgroups, Fuzzy Math. 1 (1981) 21-30.
246. Yu Chunhai, Fuzzy maximal ideals of fuzzy rings, Fuzzy Math. 6 (1 ) (1986) 21 - 23.
247. Yu Yandong, Fuzzy linear spaces redefined, Fuzzy Math. 4 (3) (1984) 59 -62.
248. Yu Yandong, Finitely generated fuzzy linear spaces, Fuzzy Math. 6 (4) (1986) 83 - 88.
249. Yu Yandong, Finitely generated T-fuzzy linear spaces, FSS 30 (1989) 69 - 8 1 .
250. Yu Yandong, Mordeson, J. N., and Cheng, Shih-Chuan, Elements of L-Algebra, Lecture Notes in Fuzzy Mathematics and Computer Science 1, Center for Research in Fuzzy Mathematics and Computer Science, Creighton University 1994.
251. Yu, Y. -D. and Wang, Z. D., TL-subrings and TL-ideals, Part 1: Basic concepts, FSS 68 (1994) 93-103.
252. Yu, Y. -D. and Wang, Z. D., TL-subrings and TL-ideals, Part 4: Radicals of TL-ideals, FSS 88 (1997) 227-236.
253. Yu, Y. -D. and Wang, Z. D., TL-subrings and TL-ideals, Part 6: Primary TL-ideals and semiprimary TL-ideals, FSS, submitted .
254. Zadeh, L. A., Fuzzy sets, Inform, and Contr. 8 (1965) 338 - 353.
255. Zahedi, M. M., L-fuzzy extended and contracted ideals, BUSEFAL 41 (1989) 104 - 110.
256. Zahedi, M. M., A characterization of L-fuzzy prime ideals, FSS 4A (1991) 147 - 160.
257. Zahedi, M. M., A note on L-fuzzy primary and semiprime ideals, FSS 51 (1992) 243 - 247.
258. Zahedi, M. M., On L-fuzzy residual quotient modules and 7r-primary submodules, FSS 51 (1992) 333 - 344.
Fuz
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/08/
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259. Zahedi, M. M., Some results on L-fuzzy modules, FSS 55 (1993) 355 -361.
260. Zahedi, M. M. and Ameri R., On fuzzy projective and injective modules, J. Fuzzy Math. 3 (1995) 181-190.
261. Zariski, O. and Samuel, P., Commutative Algebra Vol. /, D. Van Nos-trand Company, Inc., Princeton, New Jersey (1958).
262. Zhang Yue, Prime L-fuzzy ideals and primary L-fuzzy ideals, Fuzzy Math. 7 (1) (1987) 45 - 48.
263. Zhang Yue, Prime L-fuzzy ideals and primary L-fuzzy ideals, FSS 27 (1988) 345 - 350.
264. Zhang Yue and Peng Xiantu, Maximal fuzzy ideals and primary fuzzy ideals of a ring, Fuzzy Math. 4 (1) (1984) 115 - 116.
265. Zhang Yunjie and Zou Kaiqi, Normal fuzzy subgroups and conjugate fuzzy subgroups, J. Fuzzy Math. 1 (1993) 571 - 585.
266. Zhu, N., Fuzzy subgroup chains of a group and fuzzy groups, Fuzzy Math. 6(3) (1986) 7-11.
267. Zou K., Fuzzy ideals of a ring, J. Dalian Marine College 8 (2) (1982) 66 - 7 1 .
268. Zou, K., Fuzzy modules and subspaces of fuzzy modules, J. Henan Teachers University (Nat. Sci. Ed.) 2 (1983) 34-43.
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Index A a-cut, 3 a-level set, 3 Abelian, 357
L-subgroup, 20 L-subset, 10
algebraic, 166 algebraic L-variety
irreducible, 275 reducible, 275
algebraic closure, 166 algebraic L-variety, 267 algebraically independent, 167, 325 associated primes, 215
B B-linearly independent, 142 basic L-subgroup, 61 basis, 143, 236, 317, 354
c canonical representation, 347 carrier of an L-subset, 347 Cartesian cross-product, 94 Cauchy sequence, 240 characteristic function, 2 closed under projections, 34 coefficient field, 235 comaximal L-ideals, 83 compatible, 170, 192, 333 complete, 240
L-direct sum, 95 complete direct product, 4, 20-21 completion, 257 composite, 175 Condition G\ on cr, 367
Condition GT. on /x, 368 conjugate L-subgroup, 11 contracted ideal, 233 converge, 240 coprime L-ideals, 83 Correspondence Theorem, 86 coset, 12
left, 12 right, 12
crisp power set, 30
D direct product, 22, 26
complete, 20-21 weak, 22, 26
distinguished intermediate field, 192 divisible L-subgroup, 50
E exponent of inseparability, 198 extended ideal, 233 Extension Principle, 4
F /-invariant, 97 factor L-subgroup, 15, 17 factor group, 14 finite L-subset, 2 finitely generated, 123 fractionary fuzzy ideal, 263
extended integral, 265 integral, 264 invertible, 263 maximal, 264 prime, 264
free, 69, 316, 354
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maximally, 317 free L-submodule, 140 fuzzy
power set, 2 subset, 2
fuzzy C subset, 36 fuzzy chain subfield, 383 fuzzy dimension, 362 fuzzy Galois, 383 fuzzy graph, 301 fuzzy index, 363 fuzzy invertible ideal, 263 Fuzzy Luroth's Theorem, 204 fuzzy order, 363 fuzzy power set, 2 fuzzy prime spectrum, 262 fuzzy primitive, 383 fuzzy principal, 129 fuzzy relational equation, 303 fuzzy simple, 383 fuzzy subfield, 161 fuzzy subGalois, 383 fuzzy subgraph
Cartesian product of, 302 fuzzy subgroup, 7
normal, 10 fuzzy submodule, 134 fuzzy subnormal, 383 fuzzy subset, 2
G Galois, 358, 370
group, 358 over a field, 358
generate, 9, 76, 121, 236 generating set, 42, 283
5-minimal, 288 minimal, 42, 57, 283
generic point, 226 group
factor, 14 quotient, 14
group L-subalgebra, 374
H Heyting algebra, 1
homomorphic, 17, 89, 137 weakly, 89, 137
homomorphism, 17, 137 weak, 17, 89, 137
I image, 4 increasing monotonic limit, 335 index, 39, 357 inf property, 216 infinite L-subset, 2 inseparable, 171 inseparable /-algebraic, 172 intersection of L-subsets, 2 inverse image , 4 inverse of an L-subset, 6 irreducible
L-ideal, 98 element, 98
irreducible topological space, 226 irredundant representation, 142 isolated component, 215 isolated primary L-submodule, 159 isomorphic, 17, 89, 137
weakly, 89, 137 isomorphism, 17, 89, 137
weak, 17, 89, 137
L /-algebraic, 168 L-basis, 335 L-coefficient field, 243 L-contraction, 233 L-coset
left, 59 L-decomposition, 210 L-direct sum
complete, 95 external weak, 95 weak, 90
L-extension by radicals, 361 L-field
/-algebraic extension, 168 /-purely inseparable extension, 172 algebraic closure, 166 algebraic extension, 166
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algebraically closed, 166 compatible, 170 distinguished //-intermediate ex
tension, 192 extension, 161 inseparable /-algebraic extension,
172 inseparable algebraic extension, 171 modular extension, 188 neutral extension, 166 regular extension, 379 separable, 181 separable /-algebraic extension, 172 separable algebraic extension, 171 transcendental extension, 166
L-ideal, 72, 87 ^-primary, 205 £-prime, 205 ^-primary, 208 associated prime, 157, 208 comaximal, 83 contracted, 233 coprime, 83 equivalent, 284 finitely generated, 123 generalized maximal, 97 generated by, 76 irreducible, 98 local, 284 maximal, 96 normalized, 285 of maximal chain, 285 primary, 108 prime, 100 proper, 82 R-semiprimary, 117 reducible, 98 residual quotient, 148 semiprime, 102
//-intermediate field, 161 L-linearly independent, 334
maximal system, 345 L-point, 2 L-power set, 1 /^-primary
L-representation, 217
irredundant L-representation, 217 reduced L-representation, 217
L-prime spectrum, 219 /-purely inseparable, 172 L-radical, 104 L-representation
irredundant, 210 reduced, 210
L-set of generators, 121 L-singleton, 2 L-subfield, 161
generated by, 162 L-subfield in a ring, 242 L-subgroup, 7
p-primary, 48 Abelian, 20 basic, 61 conjugate, 11 cyclic, 61 divisible, 50 factor, 15, 17 generated by, 9 normal, 10, 15 pure, 52 quotient, 15, 17 reduced, 52 solvable, 20 torsion, 49
L-submodule, 134 7r-primary, 154 basis for an, 143 free, 140 generated, 139 primary, 149 primary decomposition of, 157 residual quotient, 148
L-submonoid, 23 L-subring, 71
difference, 87 generated by, 162 localized, 286 polynomial, 253 quasi-local, 229 quotient, 87, 89
L-subsemigroup, 23 L-subset, 1
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Abelian, 10 cardinality of an, 339 carrier of an, 347 closed, 312 complete direct product of, 4 contained in, 2 contains, 2 dense, 313 difference of, 70 finite, 2 generating, 139 image of, 2 infinite, 2 intersection of, 2 inverse of, 6 negative of an, 70, 131 normal, 2 open, 235, 312 product of, 6, 70 properly contained in, 2 properly contains, 2 sum of, 70, 131 support of an, 2 union of, 2 unitary, 2
L-subspace, 316, 321 basis, 324 basis of an, 317 dimension of an, 338 finite dimensional, 319 fuzzy finite dimensional, 338 infinite dimensional, 319
L-subspaces isomorphic, 345
L-topological space, 235, 312 irreducible, 313 Noetherian, 313
L-topology, 312 neutrally closed, 313
left coset, 12 left L-coset, 59 linear space, 131 linearly dependent, 47 linearly disjoint, 177 linearly independent, 47, 323 localized L-subring, 286
localized ring, 286
M maximal, 48 maximal element, 96 maximal L-ideal, 96
generalized, 97 maximally p-independent, 64 Metatheorem, 34 metricizer, 256 minimal generating set, 42, 57, 200-
201, 283 minimal L-set of generators, 125 minimal set of generators, 321
N neutral, 166 neutral closure, 170, 312 neutrally closed, 170 normal, 356-357 normal L-subgroup, 10, 15 normal L-subset, 2 normal fuzzy subgroup, 10 normalizer, 12
p p-basic, 62 p-basis, 63, 325 p-independent, 63, 325
maximally, 64 p-primary
L-subgroup, 48 component, 49
p-pure, 62 pair wise comaximal, 106 polynomial L-subring, 253 power series, 254 power set
crisp, 30 pre-image, 4 primary
L-ideal, 108 L-submodule, 149
primary decomposition irredundant, 157 normal, 157
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of L-submodules, 157 prime
L-ideal, 100 element, 100
prime L-ideal associated, 154 prime L-ideal divisor, 212
isolated, 212 minimal, 212
product of L-subsets, 6 proper L-ideal, 82 pseudo-Hausdorff, 236 pseudo-metric, 236 pure L-subgroup, 52 purely inseparable, 171
Q quasi-local ring, 228 quotient, 137, 357 quotient L-subgroup, 15, 17 quotient group, 14 quotient L-subring, 87 quotient ring, 84
R ^-primary, 208-209
L-components, 215 L-ideal, 115 L-representation, 210 isolated L-components, 215
^-radical, 111 7£-semiprimary L-ideal, 117 i?d/-primary, 207 reduced, 383 reduced L-subgroup, 52 V-reduced element, 303 reducible L-ideal, 98 regular, 1 relative p-basis, 200, 324 relatively p-independent, 200, 324 residual quotient, 145 right coset, 12 ring
quotient, 84
s saturated, 286
semiprime L-ideal, 102
separable algebraic, 170 separable /-algebraic, 172 solvable L-subgroup, 20 solvable by radicals, 362 space, 316 split, 193 strongly compatible, 170, 331 Subdirect Product Theorem, 34 subspace, 316 sup property, 2 support of an L-subset, 2
T torsion
L-subgroup, 49 closure, 50
torsion independent, 326 torsion-free basis, 326 transcendence basis, 168, 325 transcendental, 166
u union of L-subsets, 2 unitary L-subset, 2 upper well ordered, 335
v vector space, 131
w weak
L-direct sum, 90, 95 basis, 354 direct product, 22, 26 homomorphism, 17, 89, 137 isomorphism, 17, 89, 137 minimal generating set, 200-201 product, 22 sum, 132
weakly free, 354 homomorphic, 17, 89, 137 isomomorphic, 17 isomorphic, 89, 137
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