In this paper, fuzzy absolute value on a ring is defined. It is proved that every absolute value can induce a fuzzy absolute value. It is also proved that every fuzzy valued ring induce a fuzzy metric space and a fuzzy norm space. In addition, the notion of a Cauchy sequence and completion of a fuzzy valued ring are given and some of their properties are investigated. Finally, it is proved that every fuzzy valued ring has a completion.
BagT. and SamantaS.K., Fixed point theorems on fuzzy normed linear spaces, Information Sciences176(19) (2006), 2910–2931.
2.
BagT. and SamantaS.K., Finite dimensional fuzzy normed linear spaces, Journal of Fuzzy Mathematics11(3) (2003), 687–706.
3.
ChenG. and PhamT.T., Introduction to fuzzy sets, fuzzy logic, and fuzzy control systems, CRC Press, 2000.
4.
ChengS.C. and MordesonJ.N., Fuzzy linear operators and fuzzy normed linear spaces, Bull Cal Math Soc86(5) (1994), 429–436.
5.
DavvazB. and CristeaI., Fuzzy algebraic hyperstructures: An Introduction, Springer, 2015.
6.
DengZ., Fuzzy pseudo-metric spaces, J Math Anal Appl86 (1982), 74–95.
7.
EnglerA.J. and PrestelA., Valued fields, Springer, 2005.
8.
ErcegM.A., Metric spaces in fuzzy set theory, J Math Anal Appl69 (1979), 205–230.
9.
FelbinC., Finite dimensional fuzzy normed linear space, Fuzzy Sets and Systems48 (1992), 239–248.
10.
GeorgeA. and VeeramaniP., On some results in fuzzy metric spaces, Fuzzy Sets and Systems64 (1994), 395–399.
11.
GregoriV., MinanaJ., MorillasS. and SapenaA., Cauchyness and convergence in fuzzy metric spaces, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2016, pp. 1–13.
12.
KalevaO. and SeikkalaS., On fuzzy metric spaces, Fuzzy Sets and Systems12(3) (1984), 215–229.
13.
KatsarasA.K., Locally convex topologies induced by fuzzy norms, Global Journal of Mathematical Analysis1(3) (2013), 83–96.
14.
KatsarasA.K., Fuzzy topological vector spaces II, Fuzzy Sets and Systems12(2) (1984), 143–154.
15.
KramosilI. and MichálekJ., Fuzzy metrics and statistical metric spaces, Kybernetika11(5) (1975), 336–344.
16.
KrishnaS.V. and SharmaK.K.M., Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems63(2) (1994), 207–217.
17.
LeeK.Y., Approximation properties in fuzzy normed spaces, Fuzzy Sets and Systems282 (2016), 115–130.
18.
LeeB.S., LeeS.J. and ParkK.M., The completions of fuzzy metric spaces and fuzzy normed linear spaces, Fuzzy Sets and Systems106(3) (1999), 469–473.
19.
MordesonJ.N. and MalikD.S., Fuzzy commutative algebra, World Scientific, 1998.
20.
NădăbanS., Fuzzy continuous mappings in fuzzy normed linear spaces, International Journal of Computers Communications and Control10(6) (2015), 834–842.
21.
RanoG. and BagT., Fuzzy normed linear spaces, International Journal of Mathematics and Scientific Computing2(2) (2012), 16–19.
22.
RhieG.S., ChoiB.M. and DongS.K., On the completeness of fuzzy normed linear spaces, Math Japonica45(1) (1997), 33–37.
23.
SadeqiI. and AzariF.Y., The comparative study of gradual and fuzzy normed linear spaces, Journal of Intelligent and Fuzzy Systems30(2) (2016), 1195–1198.
24.
SahaP., ChoudhuryB.S. and DasP., A new contractive mapping principle in fuzzy metric spaces, Bollettino dell’Unione Matematica Italiana8(4) (2016), 287–296.
TanveerM., Note on some recent fixed point theorems in fuzzy metric spaces, Journal of Intelligent and Fuzzy Systems26(2) (2014), 811–814.
27.
TiradoP., On compactness and G-completeness in fuzzy metric spaces, Iranian Journal of Fuzzy Systems9(4) (2012), 151–158.
28.
WangL.X., A Course in fuzzy systems and control, Prentice Hall PTR, 1997.
29.
MingL.Y. and KangL.M., Fuzzy topology, World Scientific Pub, 1997.
30.
YadavD.S., ThakurS.S. and ThakurS.S., Some related fixed point theorems for four fuzzy metric spaces, Journal of Fuzzy Set Valued Analysis1 (2015), 78–85.
31.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
