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Soft substructures of rings, fields and modules. (English) Zbl 1217.16041

Summary: Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, we introduce and study soft subrings and soft ideals of a ring by using Molodtsov’s definition of the soft sets. Moreover, we introduce soft subfields of a field and soft submodule of a left \(R\)-module. Some related properties about soft substructures of rings, fields and modules are investigated and illustrated by many examples.

MSC:

16Y99 Generalizations
03E72 Theory of fuzzy sets, etc.
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[1] Zadeh, L. A., Fuzzy sets, Inform. Control, 8, 338-353 (1965) · Zbl 0139.24606
[2] Zadeh, L. A., Toward a generalized theory of uncertainty (GTU)-an outline, Inform. Sci., 172, 1-40 (2005) · Zbl 1074.94021
[3] Pawlak, Z., Rough sets, Int. J. Inform. Comput. Sci., 11, 341-356 (1982) · Zbl 0501.68053
[4] Pawlak, Z.; Skowron, A., Rudiments of rough sets, Inform. Sci., 177, 3-27 (2007) · Zbl 1142.68549
[5] Gau, W. L.; Buehrer, D. J., Vague sets, IEEE Trans. Syst. Man Cybern., 23, 2, 610-614 (1993) · Zbl 0782.04008
[6] Gorzalzany, M. B., A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems, 21, 1-17 (1987) · Zbl 0635.68103
[7] Molodtsov, D., Soft set theory-first results, Comput. Math. Appl., 37, 19-31 (1999) · Zbl 0936.03049
[8] Maji, P. K.; Roy, A. R.; Biswas, R., An application of soft sets in a decision making problem, Comput. Math. Appl., 44, 1077-1083 (2002) · Zbl 1044.90042
[9] Maji, P. K.; Biswas, R.; Roy, A. R., Soft set theory, Comput. Math. Appl., 45, 555-562 (2003) · Zbl 1032.03525
[10] Aktaş, H.; Çag˜man, N., Soft sets and soft groups, Inform. Sci., 177, 2726-2735 (2007) · Zbl 1119.03050
[11] Aktaş, H.; Çag˜man, N., Soft sets and soft groups, Inform. Sci.. Inform. Sci., Inform. Sci., 177, 3, 2726-2735 (2007), (erratum) · Zbl 1119.03050
[12] Ali, M. I.; Feng, F.; Liu, X.; Min, W. K.; Shabir, M., On some new operations in soft set theory, Comput. Math. Appl., 57, 9, 1547-1553 (2009) · Zbl 1186.03068
[13] Feng, F.; Jun, Y. B.; Zhao, X., Soft semirings, Comput. Math. Appl., 56, 2621-2628 (2008) · Zbl 1165.16307
[14] Çag˜man, N.; Enginog˜lu, S., Soft matrix theory and its decision making, Comput. Math. Appl., 59, 10, 3308-3314 (2010) · Zbl 1198.15021
[15] Acar, U.; Koyuncu, F.; Tanay, B., Soft sets and soft rings, Comput. Math. Appl., 59, 11, 3458-3463 (2010) · Zbl 1197.03048
[16] A.O. Atagün, A. Sezgin, Soft near-rings (submitted for publication).; A.O. Atagün, A. Sezgin, Soft near-rings (submitted for publication).
[17] A. Sezgin, A.O. Atagün, E. Aygün, A note on soft near-rings and idealistic soft near-rings, Filomat (in press).; A. Sezgin, A.O. Atagün, E. Aygün, A note on soft near-rings and idealistic soft near-rings, Filomat (in press). · Zbl 1289.16103
[18] Rosenfeld, A., Fuzzy groups, J. Math. Anal. Appl., 35, 512-517 (1971) · Zbl 0194.05501
[19] Abou-Zaid, S., On fuzzy subnear-rings and ideals, Fuzzy Sets and Systems, 44, 139-146 (1991) · Zbl 0772.16018
[20] Davvaz, B., Fuzzy ideals of near-rings with interval-valued membership functions, J. Sci. Islam. Repub. Iran, 12, 171-175 (2001)
[21] Davvaz, B., \((\varepsilon, \varepsilon \vee q)\)-fuzzy subnear-rings and ideals, Soft Comput., 10, 206-211 (2006) · Zbl 1084.16040
[22] Kim, K. H.; Jun, Y. B., On fuzzy ideals of near-rings, Bull. Korean Math. Soc., 33, 593-601 (1996) · Zbl 0870.16026
[23] Saikia, H. K.; Barthakur, L. K., On fuzzy \(N\)-subgroups of fuzzy ideals of near-rings and near-ring groups, J. Fuzzy Math., 11, 567-580 (2003) · Zbl 1060.16052
[24] A.O. Atagün, Soft subnear-rings, soft ideals and soft \(N\); A.O. Atagün, Soft subnear-rings, soft ideals and soft \(N\)
[25] Biswas, R.; Nanda, S., Rough groups and rough subgroups, Bull. Pol. Acad. Sci. Math., 42, 251-254 (1994) · Zbl 0834.68102
[26] Bonikowaski, Z., Algebraic Structures of Rough Sets (1995), Springer-Verlag: Springer-Verlag Berlin
[27] Iwinski, T., Algebraic approach of rough sets, Bull. Pol. Acad. Sci. Math., 35, 673-683 (1987) · Zbl 0639.68125
[28] Feng, F.; Liu, X. Y.; Leoreanu-Fotea, V.; Jun, Y. B., Soft sets and soft rough sets, Inform. Sci. (2010)
[29] Feng, F.; Li, C. X.; Davvaz, B.; Irfan Ali, M., Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Comput., 14, 899-911 (2010) · Zbl 1201.03046
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