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Sharma, Poonam Kumar
- Intuitionistic Fuzzy Prime Spectrum of a Ring
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Authors
Affiliations
1 Department of Mathematics, D.A.V. College, Jalandhar, Punjab, IN
2 IKG PT University, Jalandhar, Punjab, IN
1 Department of Mathematics, D.A.V. College, Jalandhar, Punjab, IN
2 IKG PT University, Jalandhar, Punjab, IN
Source
Fuzzy Systems, Vol 9, No 8 (2017), Pagination: 167-175Abstract
In this paper, we have introduced the topological structure on the set of all intuitionistic fuzzy prime ideals of a ring. This topology is called the Zariski topology or the intuitionistic fuzzy prime spectrum of a ring. We have shown that this topology is always T0-space and is T1-space when R is a ring in which every prime ideal is maximal, but even in this case it is not T2-space. We have also studied a special subspace Y which is always compact and is connected if and only if 0 and 1 are the only idempotent in R. We have also shown that, when the ring R is Boolean ring, then the subspace Y is also T2 – space. An embedding of space X¢ onto a subspace X* = {A∈X | A is f–invariant} has been established.
Keywords
Intuitionistic Fuzzy Ideal, Intuitionistic Fuzzy (Semi-) Prime Ideal, Intuitionistic Fuzzy Maximal Ideal, Intuitionistic Fuzzy Nil Radical of a Ring, f–Invariant Intuitionistic Fuzzy Sets, Intuitionistic Fuzzy Point.
References
- K. T. Atanassov,(1986) , Intuitionistic fuzzy sets, Fuzzy Sets and Systems, Vol. 20, No. 1, pp., 87-96.
- K. T. Atanassov, (1999) ,Intuitionistic Fuzzy sets, Theory and Applications, Studies in fuzziness and soft computing, 35, Physica-Verlag, Heidelberg.
- I. Bakhadach , S. Melliani, M. Oukessou and S.L. Chadli,(2016), Intuitionistic fuzzy ideal and intuitionistic fuzzy prime ideal in a ring, Notes on Intuitionistic Fuzzy Sets, Vol. 22, no. 2 pp., 59-63.
- D.K. Basnet, (2011), Topic in intuitionistic fuzzy algebra, Lambert Academic Publishing, ISBN: 978-3-8443-9147-3
- R. Biswas, (1989), Intuitionistic fuzzy subgroups, Math. Forum, Vol. 10, pp., 37–46.
- M. Hochster, (1969), Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142, pp., 43-60.
- M. Hochster, (1971), Existence of topologies for commutative rings with identity, Duke Math. J. 38, pp., 551-554.
- K. Hur, H.K. Kang and H.K. Song, (2003), Intuitionistic fuzzy subgroup and subrings, Honam Math J. Vol. 25, No. 1, pp., 19-41.
- K. Hur, S.Y. Jang and H.W. Kang, (2005), Intuitionistic fuzzy ideal of a ring, J. Korea Soc. Math Educ. Ser. B: Pure Appl. Math., Vol. 12, No. 3, pp., 193-209.
- Y. B. Jun, M. A. Ozturk, C. H. Park, (2007), Intuitionistic nil radicals of intuitionistic fuzzy ideals and Euclidean intuitionistic fuzzy ideals in rings, Information Sciences, Vol. 177, pp., 4662–4677.
- J.K. Kohli and Rajesh Kumar (1993), Fuzzy prime spectrum of a ring-II, Fuzzy Sets and Systems, 59, pp. 223-230.
- H.V. Kumbhojkar, (1994), Spectrum of prime fuzzy ideals, Fuzzy Sets and Systems, 62, pp., 101-109.
- C. P. Lu, (1999), The Zariski topology on the Prime Spectrum of a Module, Houston J. Math, 25 (3), pp., 417-425.
- W.J. Liu, (1982), Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8, pp., 132-139.
- Rajesh Kumar, (1992), Fuzzy prime spectrum of a ring, Fuzzy Sets and Systems, 46, pp., 147-154.
- D. S. Malik and J. N. Mordeson, (1998), Fuzzy Commutative Algebra, World Scientific Publishing Co-Pvt. Ltd.
- K. Meena and K. V. Thomas, (2011), Intuitionistic L-Fuzzy Subrings, International Mathematical Forum, Vol. 6, No. 52, pp., 2561 – 2572.
- K. Meena, (2017), Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring, Advances in Fuzzy Mathematics, Vol. 12, No. 2, pp., 229-253.
- A.V. S. N. Murty and M. N. Srinivas, (2017), Equivalent Conditions for Irreducibility of Prime Spectrum of a Ring, Advances in Fuzzy Mathematics, 4, pp., 941-944.
- A. Rosenfeld, (1971), Fuzzy groups, J. Math. Anal. Appl., 35, pp., 512-571.
- P.K. Sharma, (2016), Reducibility and Complete Reducibility of intuitionistic fuzzy G-modules , Annals of Fuzzy Mathematics and Informatics Vol. 11, No. 6, pp., 885–898.
- L. A. Zadeh, (1965), Fuzzy sets, Information and Control, Vol. 8, pp., 338–353.
- Intuitionistic Fuzzy Cosmall Submodules
Abstract Views :160 |
PDF Views:2
Authors
Affiliations
1 Department of Mathematics, D.A.V. College, Jalandhar, Punjab, IN
2 IKG PT University, Jalandhar, Punjab, IN
1 Department of Mathematics, D.A.V. College, Jalandhar, Punjab, IN
2 IKG PT University, Jalandhar, Punjab, IN
Source
Fuzzy Systems, Vol 9, No 9 (2017), Pagination: 185-188Abstract
Let M be an R-module, A and B are intuitionistic fuzzy submodules of M with A⊆B. Then A is called an intuitionistic fuzzy cosmall submodule of B in M if B/A << IF M /A (=Ω(M)/A*). In this paper an attempt has been to study intuitionistic fuzzy cosmall submodules and investigate various properties of such intuitionistic fuzzy submodules. The notion of an intuitionistic fuzzy hollow module is also introduce and a relationship of this with the intuitionistic fuzzy indecomposable module and the factor module are established.Keywords
Intuitionistic Fuzzy Small (Essential) Submodule, Intuitionistic Fuzzy Indecomposable Module, Intuitionistic Fuzzy Cosmall Submodule.References
- Atanassov, K. T. (1986) Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1), 87-96.
- Atanassov, K. T. (1999) Intuitionistic Fuzzy Sets: Theory and Applications, Series Studies on Fuzziness and Soft Computing, Vol. 35, Springer Physica-Verlag, Heidelberg.
- Basnet D. K. (2002) Intuitionistic fuzzy essential submodule; Proceeding of the ‘International Symposium on Mathematics and Its Application’, pp. 113-124.
- Biswas, R. (1989) Intuitionistic fuzzy subgroup, Mathematical Forum, X, 37-46.
- Bland Paul, E. (2012) Rings and Their Modules, Deutsche National bibliothek, Germany.
- Davvaz, B., Dudek, W.A and Jun Y.B (2006), Intuitionistic fuzzy Hv-submodules, Information Science, 176, 285-300.
- Hur, K., Kang, H. W. & Song, H. K. (2003) Intuitionistic Fuzzy Subgroups and Subrings, Honam Math J., 25(1), 19-41.
- Hur, K., Jang, S. Y. & Kang, H. W. (2005) Intuitionistic Fuzzy Ideals of a Ring, Journal of the Korea Society of Mathematical Education, Series B, 12(3), 193-209.
- John, P. P. & Isaac, P. (2012) IFSM’s of an R-Module – A Study, International Mathematical Forum, 19(7), 935-943.
- Rahman, S and Saikia, H. K. (2012) some aspects of Atanassov’s intuitionistic fuzzy submodules, Int. J. Pure and Appl. Mathematics, 77(3), 369-383.
- Sharma, P. K. (2013) (α, β)-Cut of intuitionistic fuzzy modules–II, Int. J. of Mathematical Sciences and Applications, 3(1), 11-17.
- Sharma, P.K. Kaur Gagandeep, (2016) Intuitionistic fuzzy superfluous submodule, Notes on Intuitionistic Fuzzy Sets, 22(3), 34-46.
- Sharma, P. K. (2016) Reducibility and Complete Reducibility of intuitionistic fuzzy G-modules, Annals of Fuzzy Mathematics and Informatics, 11(6), 885-898.
- Wisbaure, R. (1991) Foundations of Module and Ring Theory, Gordon and Breach: Phialdelphia.
- Zadeh, L. A. (1965) Fuzzy sets, Inform. Control. 8, 338-353.
- Intuitionistic Fuzzy Small Submodule with Respect to an Arbitrary Intuitionistic Fuzzy Submodule
Abstract Views :224 |
PDF Views:1
Authors
Affiliations
1 IKG PT University, Jalandhar, Punjab, IN
2 GNDEC, Ludhiana, IN
3 Department of Mathematics, D.A.V. College, Jalandhar, Punjab, IN
1 IKG PT University, Jalandhar, Punjab, IN
2 GNDEC, Ludhiana, IN
3 Department of Mathematics, D.A.V. College, Jalandhar, Punjab, IN
Source
Fuzzy Systems, Vol 10, No 10 (2018), Pagination: 225-229Abstract
In this paper, we introduce the concept of intuitionistic fuzzy small submodule with respect to an arbitrary intuitionistic fuzzy submodule of an R-module M. We derive the condition when an intuitionistic fuzzy submodule to be a small submodule with respect to another intuitionistic fuzzy submodule with the crisp small submodule of the R-module M. It is also shown that the sum of two intuitionistic fuzzy small submodules with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy submodule with respect to the same fixed intuitionistic fuzzy submodule. This result can be extended to an arbitrary sum of intuitionistic fuzzy submodules. Further, we prove that the homomorphic image of an intuitionistic fuzzy small submodule with respect to a fixed intuitionistic fuzzy submodule is again an intuitionistic fuzzy small submodule with respect the homomorphic image of the fixed intuitionistic fuzzy submodule.
Keywords
Intuitionistic Fuzzy Submodule (IFSM), Intuitionistic Fuzzy Small Submodule (IFSSM), B-Small Submodule, Intuitionistic Fuzzy Small Submodule with Respect to another Intuitionistic Fuzzy Submodule.References
- Atanassov, K. T. (1986) Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1), 87–96.
- Atanassov, K. T. (1999) Intuitionistic Fuzzy Sets: Theory and Applications, Series Studies on Fuzziness and Soft Computing, Vol. 35, Springer Physica-Verlag, Heidelberg.
- Zadeh, L. A. (1965) Fuzzy sets, Inform. Control, 8, 338–353
- Biswas, R. (1989) Intuitionistic fuzzy subgroup, Mathematical Forum, 10, 37–46.
- Hur, K., Kang, H. W. & Song, H. K. (2003) Intuitionistic Fuzzy Subgroups and Subrings, Honam Math J., 25(1), 19–41.
- Hur, K., Jang, S. Y. & Kang, H. W. (2005) Intuitionistic Fuzzy Ideals of a Ring, Journal of the Korea Society of Mathematical Education, Series B, 12(3), 193–209.
- Davvaz, B. Dudek, W.A and Jun, Y.B. (2006), Intuitionistic fuzzy Hv-submodules, Information Science, 176, pp. 285-300.
- Basnet, D. K. (2011) Topics in Intuitionistic Fuzzy Algebra, Lambert Academic Publishing, Germany.
- Isaac, P., & John, P. P. (2011) On Intuitionistic Fuzzy Submodules of a Module, Int. J. Of Mathematical Sciences and Applications, 1(3), 1447–1454.
- Rahman, S. & Saikia, H. K. (2012) Some aspects of Atanassov’s intuitionistic fuzzy submodules, Int. J. Pure and Appl. Mathematics, 77(3), 369–383.
- Sharma, P. K. (2013) (, )-Cut of intuitionistic fuzzy modules–II, Int. J. of Mathematical Sciences and Applications, 3(1), 11–17.
- Anderson, F. W. & Fuller K. R. (1992) Rings and Categories of Modules, Second edition, Springer Verl
- Sharma, P.K., Gagandeep Kaur, (2016), Intuitionistic fuzzy superfluous submodule, Notes on Intuitionistic Fuzzy Sets, Vol. 22 (3), pp. 34-46.
- Sharma, P.K., Gagandeep Kaur, (2018), Intuitionistic fuzzy hollow submodules, Notes on Intuitionistic Fuzzy Sets, Vol.24, no. 2, pp. 25-32.
- Beyranvand R and Moradi, F. (2015), Small submodules with respect to an arbitrary submodule, Journal of Algebra and Related Topics, Vol. 3, No 2, pp. 43-51.
- Sharma, P.K., Gagandeep Kaur, (2017), Residual quotient and annihilator of intuitionistic fuzzy sets of ring and module, International Journal of Computer Science & Information Technology (IJCSIT), Vol 9, No 4, pp. 1-15.