Some topological properties of spectrum of fuzzy submodules

Document Type: Research Paper

Authors

1 School of Mathematics, Statistics and Computer Science, College of Sciences, University of Tehran, Teheran, Iran

2 Department of Mathematics, Semnan University, Semnan, Iran

Abstract

Let $R$ be a commutative ring with identity and $M$ be an
$R$-module. Let $FSpec(M)$ denotes the collection of all prime fuzzy
submodules of $M$. In this regards some basic properties of Zariski
topology on $FSpec(M)$ are investigated. In particular, we prove
some equivalent conditions for irreducible subsets of this
topological space and it is shown under certain conditions
$FSpec(M)$ is a $T_0-$space or Hausdorff.

Keywords


[1] R. Ameri, Some properties of zariski topology of multiplication modules, Houston Journal of
Mathematics, 36(2) (2009), 337-344.
[2] R. Ameri and R. Mahjoob, Prime spectrum of L-Submodules, Fuzzy Sets and Systems, 159(9)
(2008), 1107-1115.
[3] R. Ameri and R. Mahjoob, Zariski topology on the spectrum of prime L-submodules, Soft
Comput., 12(9) (2008), 901-908.
[4] S. K. Bhambri, R. Kumar and P. Kumar,Fuzzy prime submodules and radical of a fuzzy
submodules, Bull. Cal. Math. Soc., 87 (1993), 163-168.
[5] V. N. Dixit, R. Kummar and N. Ajmal,Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy
Sets and Systems, 44 (1991), 127-138.
[6] J. A. Goguen, L-fuzzy sets, Journal Math. Appl., 18 (1967) 145-174.
[7] H. Hadji-Abadi and M. M. Zahedi, Some results on fuzzy prime spectrum of a ring, Fuzzy
Sets and Systems, 77 (1996), 235-240.
[8] R. Kumar, Fuzzy prime spectrum of a ring, Fuzzy Sets and Systems, 46 (1992), 147-154.
[9] R. Kumar and J. K. Kohli,Fuzzy prime spectrum of a ring II, Fuzzy Sets and Systems, 59
(1993), 223-230.
[10] H. V. Kumbhojkar,Some comments on spectrum of prime fuzzy ideals of a ring, Fuzzy Sets
and Systems, 85 (1997), 109-114.

[11] H. V. Kumbhojkar,Spectrum of prime fuzzy ideals, Fuzzy Sets and Systems, 62 (1994), 101-
109.
[12] Chin. Pi. Lu,Prime submodules of modules, Comm. Math. Univ., 33 (1987), 61-69.
[13] Chin.Pi. Lu, The zariski topology on the spectrum of a modules, Houston Journal of Mathe-
matics, 25(3) (1999), 417-432.
[14] Chin.Pi. Lu,Spectra of modules, Comm. in Algebra, 23(10) (1995) 3741-3752.
[15] R. L. McCasland, M. E. Moore and P. F. Smith,On the Spectrum of Modules Over a Com-
mutative Ring, Communications in Algebra, 25(1) (1997), 79-103.
[16] John. N. Mordeson and D. S. Malik,Fuzzy Commutative Algebra, World Scienti c Publishing
Co. Pet. Ltd, 1998.
[17] T. K. Mukherjee and M. K. Sen,On fuzzy ideals of a ring I; Fuzzy Sets and systems, 21
(1987), 99-104.
[18] C. V. Negoita and D. A. Ralescu, Application of fuzzy systems analysis, Basel and Stuttgart,
Birkhauser Verlag; New York, Wiley- Halstead, (1975), pp. 191.
[19] F. Z. Pan, Fuzzy nitely generated modules, Fuzzy Sets and Systems, 21 (1987), 105-113.
[20] R. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.
[21] F. I. Sidky, On radical of fuzzy submodules and primary fuzzy submodules, Fuzzy Sets and
Systems, 119 (2001), 419-425.
[22] L. A. Zadeh, Fuzzy sets, Inform and Control, 8 (1965), 338-353.