Research Article
BibTex RIS Cite

Year 2022, Volume: 32 Issue: 32, 46 - 61, 16.07.2022

Sachin Sarode Vinayak Joshı

https://doi.org/10.24330/ieja.1102289

Abstract

References

  • F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract commutative ideal theory, Period. Math. Hung., 30(1) (1995), 1-26.
  • D. D. Anderson, Abstract commutative ideal theory without chain condition, Algebra Universalis, 6 (1976), 131-145.
  • R. G. Burton, Fractional elements in multiplicative lattices, Pacific J. Math., 56(1) (1975), 35-49.
  • R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math., 12 (1962), 481-498.
  • L. Fuchs and R. Reis, On lattice-ordered commutative semigroups, Algebra Universalis, 50 (2003), 341-357.
  • G. Gratzer, General Lattice Theory, Birkhauser Verlag, Basel, 1998.
  • V. Joshi and S. B. Ballal, A note on n-Baer multiplicative lattices, Southeast Asian Bull. Math., 39 (2015), 67-76.
  • C. Jayaram, Primary elements in Prufer lattices, Czechoslovak Math. J., 52(127) (2002), 585-593.
  • C. Jayaram and E. W. Johnson, s-prime elements in multiplicative lattices, Period. Math. Hungar., 31 (1995), 201-208.
  • C. Jayaram and E. W. Johnson, Strong compact elements in multiplicative lattices, Czechoslovak Math. J., 47(122) (1997), 105-112.
  • C. Jayaram, U. Tekir and E. Yetkin, 2-absorbing and weakly 2-absorbing elements in multiplicative lattices, Comm. Algebra, 42 (2014), 1-16.
  • H. A. Khashan and A. B. Bani-Ata, $J$-ideals of commutative rings, Int. Electron. J. Algebra, 29 (2021), 148-164.
  • C. S. Manjarekar and A. V. Bingi, $\phi$-prime and $\phi$-primary elements in multiplicative lattices, Algebra, (2014), 890312 (7 pp).
  • R. Mohamadian, $r$-ideals in commutative rings, Turkish J. Math., 39 (2015), 733-749.
  • U. Tekir, S. Koc and K. H. Oral, $n$-ideals of commutative rings, Filomat, 31(10) (2017), 2933-2941.
  • M. Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc., 45 (1939), 335-354.

$\mathfrak{X}$-elements in multiplicative lattices - A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings

Year 2022, Volume: 32 Issue: 32, 46 - 61, 16.07.2022

Sachin Sarode Vinayak Joshı

https://doi.org/10.24330/ieja.1102289

Abstract

In this paper, we introduce the concept of an $\mathfrak{X}$-element with respect to an $M$-closed set $\mathfrak{X}$ in multiplicative lattices and study properties of $\mathfrak{X}$-elements. For a particular $M$-closed subset $\mathfrak{X}$, we define the concepts of $r$-elements, $n$-elements and $J$-elements. These elements generalize the notion of $r$-ideals, $n$-ideals and $J$-ideals of a commutative ring with identity to multiplicative lattices. In fact, we prove that an ideal $I$ of a commutative ring $R$ with identity is a $n$-ideal ($J$-ideal) of $R$ if and only if it is an $n$-element ($J$-element) of $Id(R)$, the ideal lattice of $R$.

Keywords

$\mathfrak{X}$-element $n$-element $r$-element $n$-ideal $r$-ideal $J$-ideal $J$-element Multiplicative lattice prime element commutative ring

References

  • F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract commutative ideal theory, Period. Math. Hung., 30(1) (1995), 1-26.
  • D. D. Anderson, Abstract commutative ideal theory without chain condition, Algebra Universalis, 6 (1976), 131-145.
  • R. G. Burton, Fractional elements in multiplicative lattices, Pacific J. Math., 56(1) (1975), 35-49.
  • R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math., 12 (1962), 481-498.
  • L. Fuchs and R. Reis, On lattice-ordered commutative semigroups, Algebra Universalis, 50 (2003), 341-357.
  • G. Gratzer, General Lattice Theory, Birkhauser Verlag, Basel, 1998.
  • V. Joshi and S. B. Ballal, A note on n-Baer multiplicative lattices, Southeast Asian Bull. Math., 39 (2015), 67-76.
  • C. Jayaram, Primary elements in Prufer lattices, Czechoslovak Math. J., 52(127) (2002), 585-593.
  • C. Jayaram and E. W. Johnson, s-prime elements in multiplicative lattices, Period. Math. Hungar., 31 (1995), 201-208.
  • C. Jayaram and E. W. Johnson, Strong compact elements in multiplicative lattices, Czechoslovak Math. J., 47(122) (1997), 105-112.
  • C. Jayaram, U. Tekir and E. Yetkin, 2-absorbing and weakly 2-absorbing elements in multiplicative lattices, Comm. Algebra, 42 (2014), 1-16.
  • H. A. Khashan and A. B. Bani-Ata, $J$-ideals of commutative rings, Int. Electron. J. Algebra, 29 (2021), 148-164.
  • C. S. Manjarekar and A. V. Bingi, $\phi$-prime and $\phi$-primary elements in multiplicative lattices, Algebra, (2014), 890312 (7 pp).
  • R. Mohamadian, $r$-ideals in commutative rings, Turkish J. Math., 39 (2015), 733-749.
  • U. Tekir, S. Koc and K. H. Oral, $n$-ideals of commutative rings, Filomat, 31(10) (2017), 2933-2941.
  • M. Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc., 45 (1939), 335-354.
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Sachin Sarode This is me

Vinayak Joshı This is me

Publication Date July 16, 2022
Published in Issue Year 2022 Volume: 32 Issue: 32

Cite

APA Sarode, S., & Joshı, V. (2022). $\mathfrak{X}$-elements in multiplicative lattices - A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings. International Electronic Journal of Algebra, 32(32), 46-61. https://doi.org/10.24330/ieja.1102289
AMA Sarode S, Joshı V. $\mathfrak{X}$-elements in multiplicative lattices - A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings. IEJA. July 2022;32(32):46-61. doi:10.24330/ieja.1102289
Chicago Sarode, Sachin, and Vinayak Joshı. “$\mathfrak{X}$-Elements in Multiplicative Lattices - A Generalization of $J$-Ideals, $n$-Ideals and $r$-Ideals in Rings”. International Electronic Journal of Algebra 32, no. 32 (July 2022): 46-61. https://doi.org/10.24330/ieja.1102289.
EndNote Sarode S, Joshı V (July 1, 2022) $\mathfrak{X}$-elements in multiplicative lattices - A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings. International Electronic Journal of Algebra 32 32 46–61.
IEEE S. Sarode and V. Joshı, “$\mathfrak{X}$-elements in multiplicative lattices - A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings”, IEJA, vol. 32, no. 32, pp. 46–61, 2022, doi: 10.24330/ieja.1102289.
ISNAD Sarode, Sachin - Joshı, Vinayak. “$\mathfrak{X}$-Elements in Multiplicative Lattices - A Generalization of $J$-Ideals, $n$-Ideals and $r$-Ideals in Rings”. International Electronic Journal of Algebra 32/32 (July 2022), 46-61. https://doi.org/10.24330/ieja.1102289.
JAMA Sarode S, Joshı V. $\mathfrak{X}$-elements in multiplicative lattices - A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings. IEJA. 2022;32:46–61.
MLA Sarode, Sachin and Vinayak Joshı. “$\mathfrak{X}$-Elements in Multiplicative Lattices - A Generalization of $J$-Ideals, $n$-Ideals and $r$-Ideals in Rings”. International Electronic Journal of Algebra, vol. 32, no. 32, 2022, pp. 46-61, doi:10.24330/ieja.1102289.
Vancouver Sarode S, Joshı V. $\mathfrak{X}$-elements in multiplicative lattices - A generalization of $J$-ideals, $n$-ideals and $r$-ideals in rings. IEJA. 2022;32(32):46-61.