Let S be a multiplicative subset of a commutative ring R. We considered a R × S as a universal and introduced the notion of lower[upper] approximations in a R × S . Also, we gave some properties of the lower[upper] approximations in a R × S . We obtained some properties of rough ideals
BonikowaskiZ., Algebraic structures of rough sets, in: W.Ziarko (Ed.) Rough Sets Fuzzy Sets, and Knowledge Discovery, Springer-Verlag, Berlin, (1995), pp. 242–247
2.
BiswasR. and NandaS., Rough groups and rough subgroups, Bull Polish Acad Sci Math42 (1994), 251–254.
3.
CorsiniP., Rough sets, fuzzy sets and join spaces, Honorary volume dedicated to Prof. Emeritus J. Mittas, Aristotle University of Thessaloniki, 1999–2000.
4.
ComerS.D., On connections between information systems rough, sets and algebraic logic, Algebraic Methods Logic Compuct. Sci., 28, Banach Center Publications, 1993, pp. 117–124.
5.
DavvazB., Roughness based on fuzzy ideals, Inform Sci176 (2006), 2417–2437.
6.
DavvazB., Roughness in rings, Inform Sci164 (2004), 147–163.
7.
DavvazB., Rough sets in a fundamental ring, Bull Iranian Math Soc24 (1998), 49–61.
8.
DavvazB. and MahdavipourM., Roughness in modules, Inform Sci176 (2006), 3658–3674.
9.
DuboisD. and PradeH., Rough fuzzy sets and fuzzy rough sets, Int J General Syst17(2-3) (1990), 191–209.
10.
IwinskiT. and PradeH., Algebraic approach to rough sets, Bull Polish Acad Sci Math35 (1987), 673–683.
KurokiN., Rough ideals in semigroups, Inform Sci100 (1997), 139–163.
13.
KurokiN. and MordesonJ.N., Structure of rough sets and rough groups, J Fuzzy Math5(1) (1997), 183–191.
14.
KurokiN. and WangP.P., The lower and upper approximations in a fuzzy group, Inform Sci90 (1996), 203–220.
15.
MordesonJ.N., Rough set theory applied to (fuzzy) ideal theory, Fuzzy Sets and Systems121 (2001), 315–324.
16.
MukherjeeT.K. and SenM.K., On fuzzy ideals in rings, Fuzzy Sets Syst21 (1987), 99–104.
17.
PawlakZ., Rough sets, Int J Comput Inform Sci11 (1982), 341–356.
18.
PawlakZ., Rough Sets-Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991.
19.
PawlakZ. and SkowronA., Rudiments of rough sets, Inform Sci177(1) (2007), 3–27.
20.
PawlakZ. and SkowronA., Rough sets: Some extensions, Inform Sci177(1) (2007), 28–40.
21.
PawlakZ. and SkowronA., Rough sets and Boolean reasoning, Inform Sci177(1) (2007), 41–73.
22.
PomykalaJ. and PomykalaJ.A., The stone algebra of rough sets, Bull Polish Acad Sci Math36 (1988), 495–508.
23.
SarkarM., Rough-fuzzy functions in classification, Fuzzy Sets Syst132 (2002), 353–369.
24.
LiuW., Operations on fuzzy ideals, Fuzzy Sets and Systems8 (1983), 31–41.
25.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
26.
ZadehL.A., The concept of linguistic variable and its applications to approximate reasoning, Part I, Inform Sci8 (1975), 199–249; Part II, Inform Sci8 (1975), 301–357; Part II, Inform Sci9 (1976), 43–80.
