On essential and closed Fuzzy ideals of a semiring

Diksha Patwari; Nabanita Goswami; Helen K. Saikia

doi:10.5269/bspm.65238

Diksha Patwari Gauhati University
Nabanita Goswami "Gauhati University"
Helen K. Saikia Gauhati University

Abstract

In this paper, existence of essential fuzzy ideal of a semiring S is shown and some of the properties of these ideals are investigated. The study of essentiality helps to develop the concepts like closed fuzzy ideal and relative fuzzy complement of S. Fuzzy ideal µ of S ( or δ) is said to be closed ideal of S ( or δ) if µ has no non-constant (proper) essential extension. Various results on closed fuzzy ideal of S are studied. Also relation between complementary summand and relative complement is established.

Downloads

Download data is not yet available.

Author Biography

Helen K. Saikia, Gauhati University

Professor, Department of Mathematics

References

A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl, 35, 512-517, (1971).

C. V. Negotia, D. A. Ralescu, Application of Fuzzy Sets to System, Birkhauser Verlag: Basel and Stuttgart, Germany, (1975).

D. S. Malik, Fuzzy ideals of artinian, Fuzzy sets and systems, 46, 111-115, (1990).

F. Pan, Fuzzy finitely generated modules, Fuzzy Sets and Systems, 21, 105-113, (1987).

F. I. Sidki,On radicals of fuzzy submodules and primary fuzzy submodules, Fuzzy Sets and System, 119, 419-425, (2001).

G. J. Klir, B.Yuan, Fuzzy Sets and Fuzzy Logic, Theory and Application.Prentice Hall P T R: New Jersey, (1995).

H. S. Vandiver, Note On a Simple Type of Algebra in which the Cancellation Law of Addition Does Not Hold, Bulletin of the American Mathematical Society, 40, 914-920, (1934).

J. Ahsan, J. N. Mordeson, M. Shabir, Fuzzy Semirings with Application to Automata Theory, Springer-Verlag: Berlin Heidelberg, (2012).

J. N. Mordeson, D. S. Malik, Fuzzy Commutative Algebra, World Scientific Publishing Co. Pte. Ltd.: Singapore, (1998).

J. S. Golan, Semirings and Their Applications, Netherlands: Springer, (1999).

J. Zhan, Z. Tan, T-fuzzy k-ideals of semirings, Sci. Math. Japon, 58, 597-601, (2003).

K. H. Kim and Y. B. Jun, A note on fuzzy R-subgroups of near-rings, Soochow Journal of Mathematics, 28, 339–346, (2002).

L. Wangjin, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and System, 8, 133-139, (1982).

L. Wangjin, Fuzzy prime ideals and fuzzy radical ideals, Inform. Sci, 46, 1-12, (1990).

L. A. Zadeh, Fuzzy Sets, Inform, and Contr, 8, 338-353, (1965).

M. C. Kalita. A Study of Fuzzy Algebraic Structures: Some Special Types [Doctoral dissertation, Gauhati University], Shodhganga :a reservoir of Indian theses,(2007).

M. C. Kalita, H. K. Saikia, On fuzzy essential submodules, Journal of Fuzzy Mathematics, 17, 109–117, (2009).

M. C. Kalita, H. K. Saikia, On annihilator of fuzzy subset of modules, International Journal of Algebra, 3, 483-488, (2009).

M. C. Kalita, H. K. Saikia, Singular fuzzy ideals of a commutative ring. Journal of Intelligent and Fuzzy System, 30, 3543-3549, (2006).

R. Kumar, Certain fuzzy ideals of rings redefined, Fuzzy sets and systems, 46, 251–260, (1992).

R. Kumar, Fuzzy ideals and fuzzy semi-prime ideals, Some ring theoretic analogues, Information Sci, 46, 43–52, (1992).

S. Abou-Zaid, On fuzzy subnear-rings and ideals, Fuzzy Sets and Systems, 44, 139–146, (1991).

S. D. Kim and H. S. Kim, On fuzzy ideals of near-rings, Bulletin of the Korean Mathematical Society, 33, 593–601,(1996)

S. Ghosh, Fuzzy k-ideals of semirings, Fuzzy Sets and Systems, 95, 103-108, (1998).

S. Rahman, H. K. Saikia, On the definition of intuitionistic fuzzy h-ideals of hemirings, Kyungpook Mathematical Journal, 53, 435-457, (2013).

T. K. Mukherjee, M. K. Sen and D. Roy, On fuzzy submodules of their radicals, J. Fuzzy Math, 4, 549-558, (1996).

T. K. Mukherjee, M. K. Sen, On fuzzy ideals of a ring I, Fuzzy Sets and Systems, 21, 99-104, (1987).

U. Medhi, H. K. Saikia, B. Davvaz, An investigation of F-Closure of fuzzy submodules of a module, Fuzzy information and engineering, 11, 212-220, (2019).

M. M. Zahedi, Some results on L-fuzzy modules, Fuzzy Sets and System, 55, 355-361, (1993).