On Generalization of $\delta$-Primary Elements in Multiplicative Lattices

Ashok V. Bingi1

1Department of Mathematics, St. Xavier's College (Autonomous), Mumbai, Maharashtra, India.

Abstract: In this paper, we introduce $\phi$-$\delta$-primary elements in a compactly generated multiplicative lattice $L$ and obtain its characterizations. We prove many of its properties and investigate the relations between these structures. By a counter example, it is shown that a $\phi$-$\delta$-primary element of $L$ need not be $\delta$-primary and found conditions under which a $\phi$-$\delta$-primary element of $L$ is $\delta$-primary.
Keywords: Expansion function, $\delta$-primary element, $\phi$-$\delta$-primary element, 2-potent $\delta$-primary element, $n$-potent $\delta$-primary element, global property.

Cite this article as: Ashok V. Bingi, On Generalization of $\delta$-Primary Elements in Multiplicative Lattices, Int. J. Math. And Appl., vol. 9, no. 2, 2021, pp. 69-80.

  1. F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract commutative ideal theory, Periodica Mathematica Hungarica, 30(1)(1995), 1-26.
  2. D. D. Anderson and M. Bataineh, Generalizations of prime ideals, Communications in Algebra, 36(2)(2008), 686-696.
  3. D. S. Culhan, Associated primes and primal decomposition in modules and lattice modules and their duals, Ph. D. Thesis, University of California, Riverside, (2005).
  4. A. Y. Darani, Generalizations of primary ideals in commutative rings, Novi Sad J. Math., 42(1)(2012), 27-35.
  5. R. P. Dilworth, Abstract commutative ideal theory, Pacific Journal of Mathematics, 12(2)(1962), 481-498.
  6. M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Communications in Algebra, 40(4)(2012), 1268-1279.
  7. A. Jaber, Properties of $\phi$-$\delta$-primary and 2-absorbing $\delta$-primary ideals of commutative rings, Asian-European Journal of Mathematics, 13(1)(2020), 1-11.
  8. C. S. Manjarekar and A. V. Bingi, $\delta$-primary elements in multiplicative lattices, International Journal of Advance Research, 2(6)(2014), 1-7.
  9. C. S. Manjarekar and A. V. Bingi, $\phi$-Prime and $\phi$-Primary Elements in Multiplicative Lattices, Algebra, (2014), 1-7.
  10. N. K. Thakare and C. S. Manjarekar, Radicals and uniqueness theorem in multiplicative lattices with chain conditions, Studia Scientifica Mathematicarum Hungarica, 18(1983), 13-19.
  11. J. Wells, The restricted cancellation law in a Noether lattice, Fundamenta Mathematicae, 3(75)(1972), 235-247.
  12. D. Zhao, $\delta$-primary ideals of commutative rings, Kyungpook Math. J, 41(2001), 17-22.