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Fuzzy Information and Engineering 2023, 15(4): 313-323 https://doi.org/10.26599/FIE.2023.9270022
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Prime Ideal, Semiprime Ideal, and Radical of an Ideal of an L-Subring

Show Author's Information Anand Swaroop Prajapati1,  Naseem Ajmal2,  Iffat Jahan3( )
Atma Ram Sanatan Dharma College, University of Delhi, New Delhi 110021, India
Zakir Husain College, University of Delhi, New Delhi 110006, India
Ramjas College, University of Delhi, New Delhi 110007, India
Keywords:
prime ideal, semiprime ideal, radical of an ideal, L-subring
Cite this article:
Prajapati AS, Ajmal N, Jahan I. Prime Ideal, Semiprime Ideal, and Radical of an Ideal of an L-Subring. Fuzzy Information and Engineering, 2023, 15(4): 313-323. https://doi.org/10.26599/FIE.2023.9270022
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Abstract Full text About this article

In this paper, we develop a systematic theory for the ideals of an L-ring L(μ, R). We introduce the concepts of a prime ideal, a semiprime ideal, and the radical of an ideal in an L-ring. The notion of a maximal ideal has been introduced and discussed in different studies. We prove several results pertaining to these notions which are versions of their counterparts in classical ring theory. Besides this, we prove that for a commutative ring R, the radical η of an ideal η in an L-ring L(μ, R) is an ideal of μ provided that η has sup-property.


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Prime Ideal, Semiprime Ideal, and Radical of an Ideal of an L-Subring

Show Author's information Anand Swaroop Prajapati1, Naseem Ajmal2, Iffat Jahan3( )
Atma Ram Sanatan Dharma College, University of Delhi, New Delhi 110021, India
Zakir Husain College, University of Delhi, New Delhi 110006, India
Ramjas College, University of Delhi, New Delhi 110007, India

Abstract

In this paper, we develop a systematic theory for the ideals of an L-ring L(μ, R). We introduce the concepts of a prime ideal, a semiprime ideal, and the radical of an ideal in an L-ring. The notion of a maximal ideal has been introduced and discussed in different studies. We prove several results pertaining to these notions which are versions of their counterparts in classical ring theory. Besides this, we prove that for a commutative ring R, the radical η of an ideal η in an L-ring L(μ, R) is an ideal of μ provided that η has sup-property.

Keywords: prime ideal, semiprime ideal, radical of an ideal, L-subring

References(16)

[1]

A. S. Prajapati and N. Ajmal, Maximal ideals of L-subring, J. Fuzzy Math., vol. 15, pp. 383–398, 2007.
Google Scholar
[2]

A. S. Prajapati and N. Ajmal, Maximal ideals of L-subring II, J. Fuzzy Math., vol. 15, pp. 399–411, 2007.
Google Scholar
[3]

A. S. Prajapati, Residual of ideals of an L-ring, Iran. J. Fuzzy Syst., vol. 4, no. 2, pp. 69–82, 2007.
Google Scholar
[4]
A. S. Prajapati, N. Ajmal, and I. Jahan, Radical of an ideal and primary ideal of an L-subring, http://www.afmi.or.kr, 2023.
[5]
M. M. Glukhov, I. V. Stelletskii, and T. S. Fofanova, Lattice theory, in Progress in Mathematics, A. Chambert-Loir, J. H. Lu, M. Ruzhansky, and Y. Tschinkel, eds. Boston, MA, USA: Springer, 1972, pp. 111–170.
DOI
[6]
J. N. Mordeson, L-subspaces and L-subfield, in Lecture Notes in Fuzzy Mathematics and Computer Science, Y. D. Yang, J. N. Mordeson, and S. C. Cheng, eds. Omaha, NE, USA: Creighton University, 1996.
[7]
J. N. Mordeson and D. S. Malik, Fuzzy Commutative Algebra. Singapore: World Scientific, 1998.
DOI
[8]
G. Szasz, Introduction to Lattice Theory. New York, NY, USA: Academic Press, 1963.
[9]
Y. Yandong, J. N. Mordeson, and S. C. Cheng, Elements of L-Algebra (Lecture Notes in Fuzzy Mathematics and Computer Science), Omaha, NE, USA: Creighton University, 1994.
[10]

A. S. Malik and J. N. Mordeson, Fuzzy prime ideals of a ring, Fuzzy Sets Syst., vol. 37, no. 1, pp. 93–98, 1990.
DOI Google Scholar
[11]

D. S. Malik and J. N. Mordeson, Fuzzy maximal, radical and primary ideals of a ring, Inf. Sci., vol. 53, no. 3, pp. 237–250, 1991.
DOI Google Scholar
[12]

D. S. Malik and J. N. Mordeson, Fuzzy primary representations of fuzzy ideals, Inf. Sci., vol. 55, no. 1–3, pp. 151–165, 1991.
DOI Google Scholar
[13]

D. S. Malik and J. N. Mordeson, Radicals of fuzzy ideals, Inf. Sci., vol. 65, no. 3, pp. 239–252, 1992.
DOI Google Scholar
[14]

D. S. Malik and J. N. Mordeson, R-primary representation of L-ideals, Inform. Sci., vol. 88, pp. 227–246, 1996.
DOI Google Scholar
[15]
N. Ajmal and A. S. Prajapati, Prime radical and primary decomposition of ideals in an L-subring, Annal. Fuzzy Math. Inform., vol. 25, no. 3, pp. 293–302, 2023.
[16]
E. M. Patterson, A first course in rings and ideals, The Mathematical Gazette, vol. 56, no. 396, pp. 170–171, 1972.
DOI
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Received: 23 December 2022
Revised: 21 February 2023
Accepted: 26 March 2023
Published: 02 January 2024
Issue date: December 2023

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© The Author(s) 2023. Published by Tsinghua University Press.

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This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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