Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On 2-absorbing Primary Ideals of Commutative Semigroups

  • Received : 2021.01.05
  • Accepted : 2022.05.03
  • Published : 2022.09.30
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Abstract

In this paper we introduce the notion of 2-absorbing primary ideals of a commutative semigroup. We establish the relations between 2-absorbing primary ideals and prime, maximal, semiprimary and 2-absorbing ideals. We obtain various characterization theorems for commutative semigroups in which 2-absorbing primary ideals are prime, maximal, semiprimary and 2-absorbing ideals. We also study some other important properties of 2-absorbing primary ideals of a commutative semigroup.

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Acknowledgement

The authors express their deep gratitude to learned referees for their meticulous reading and valuable suggestions which have definitely improved the paper. The research work reported here is supported by UGC-JRF NET Fellowship (Award No. 11-04-2016-421922) to Biswaranjan Khanra by the University Grants Commission, Governtment of India.

References

  1. D. D. Anderson and E. W. Johnson, Ideal theory in commutative semigroup, Semigroup Forum, 30(2)(1984), 127-158. https://doi.org/10.1007/BF02573445
  2. A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75(3)(2007), 417-429. https://doi.org/10.1017/S0004972700039344
  3. A. Badawi and U. Tekir, On 2-absorbing primary ideals in commutative ring, Bull. Korean. Math. Soc., 51(4)(2014), 1163-1173. https://doi.org/10.4134/BKMS.2014.51.4.1163
  4. H. Cay, H. Mostafanasab, G. Ulucak and U. Tekir, On 2-absorbing and strongly 2-absorbing ideals of commutative semigroups, Annale Stiintifice ale Universitatii Al I Cuza din lasi-Mathematica Vol. 3 (2016), 871-881.
  5. M. A. Erbay, U. Tekir and S. Koc, r-ideals of commutative semigroups, Int. J. Algebra, 10(11)(2016), 525-533. https://doi.org/10.12988/ija.2016.61276
  6. B. Khanra and M. Mandal, Semigroups in which 2-absorbing ideals are prime and maximal, Quasigroups Related Systems, 29(2)(2021), 213-222.
  7. J. Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc., 13(3)(1963) 31-50. https://doi.org/10.1112/plms/s3-13.1.31
  8. P. Kumar, M. K. Dubey and P. Sarohe, On 2-absorbing primary ideals in commutative semiring, Eur. J. Pure Appl. Math., 9(2)(2016), 186-195.
  9. H. Lal, Commutative semi-primary semigroups, Czechoslovak Math. J., 25(100)(1975), 1-3. https://doi.org/10.21136/CMJ.1975.101286
  10. H. Mostafanasab and A. H. Darani, Some properties of 2-absorbing and weakly 2-absorbing primary ideals, Transactions on Algebra and its Application, 1(1)(2015), 10-18.
  11. M. Satyanarayana, Commutative primary semigroups, Czechoslovak Math. J., 22(4)(1972), 509-516. https://doi.org/10.21136/cmj.1972.101121
  12. M. Satyanarayana, Commutative semigroups in which primary ideals are prime, Math. Nachr., 48(1971), 107-111. https://doi.org/10.1002/mana.19710480108
  13. S. Schwartz, prime and maximal ideals of a semigroup, Czechoslovak Mathematical Journal.
  14. F. Soheilinia, On 2-absorbing and weakly 2-absorbing primary ideals of commutative semiring, Kyungpook Math. J. 56(2016), 107-120. https://doi.org/10.5666/KMJ.2016.56.1.107
  15. M. P. Wasadikar and K. T. Gaikwad, Some properties of 2-absorbing primary ideals in lattices, AKCE Int. J. Graphs Comb., 16(2019) 18-26. https://doi.org/10.1016/j.akcej.2018.01.015