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Abstract
Let £ be a bounded distributive lattice and S a join-subset of £. In this paper, we introduce the concept of S-prime elements (resp. weakly S-prime elements) of £. Let p be an element of £ with S ∧p = 0 (i.e. s∧p = 0 for all s ∈ S). We say that p is an S-prime element (resp. a weakly S-prime element) of £ if there is an element s ∈ S such that for all x, y ∈ £ if p ≤ x ∨ y (resp. p ≤ x ∨ y 6= 1), then p ≤ x ∨ s or p ≤ y ∨ s. We extend the notion of S-prime property in commutative rings to S-prime property in lattices.
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References
- Anderson D. D. and Smith E., Weakly prime ideals, Houston J. Math., 29 (4) (2003), 831-840.
- Aqalmoun M., S-prime ideals in principal domain, J. Indones. Math. Soc., 29 (1) (2023), 93-98.
- Almahdi F. A. A., Bouba E. M. and Tamekkante M., On weakly S-prime ideals of commutative rings, An. St. Univ. Ovidius Constanta, 29 (2) (2021), 173-186.
- Birkhoff G., Lattice theory, Amer. Math. Soc., (1973). DOI: 10.1090/coll/025.
- Badawi A., On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75 (3) (2007), 417-429.
- C˘alug˘areanu G., Lattice Concepts of Module Theory, Kluwer Academic Publishers, 2000. DOI: 10.1007/978-94-015-9588-9.
- Ebrahimi Atani S. and sedghi Shanbeh Bazari M., On 2-absorbing filters of lattices, Discuss. Math. Gen. Algebra Appl., 36 (2016), 157-168. DOI: 10.7151/dmgaa.1253.
- Ebrahimi Atani S. and Farzalipour F., On weakly primary ideals, Georgian Mathematical Journal , 12 (2005), 423-429.
- Hamed A. and Malek A., S-prime ideals of a commutative ring, Beitr. Algebra Geom. (2019).
- Sevim E. S., Arabaci T., Tekir U. and Koc S., On ¨ S-prime submodules, Turk. J. Math., 43(2) (2019), 1036-1046.
References
Anderson D. D. and Smith E., Weakly prime ideals, Houston J. Math., 29 (4) (2003), 831-840.
Aqalmoun M., S-prime ideals in principal domain, J. Indones. Math. Soc., 29 (1) (2023), 93-98.
Almahdi F. A. A., Bouba E. M. and Tamekkante M., On weakly S-prime ideals of commutative rings, An. St. Univ. Ovidius Constanta, 29 (2) (2021), 173-186.
Birkhoff G., Lattice theory, Amer. Math. Soc., (1973). DOI: 10.1090/coll/025.
Badawi A., On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75 (3) (2007), 417-429.
C˘alug˘areanu G., Lattice Concepts of Module Theory, Kluwer Academic Publishers, 2000. DOI: 10.1007/978-94-015-9588-9.
Ebrahimi Atani S. and sedghi Shanbeh Bazari M., On 2-absorbing filters of lattices, Discuss. Math. Gen. Algebra Appl., 36 (2016), 157-168. DOI: 10.7151/dmgaa.1253.
Ebrahimi Atani S. and Farzalipour F., On weakly primary ideals, Georgian Mathematical Journal , 12 (2005), 423-429.
Hamed A. and Malek A., S-prime ideals of a commutative ring, Beitr. Algebra Geom. (2019).
Sevim E. S., Arabaci T., Tekir U. and Koc S., On ¨ S-prime submodules, Turk. J. Math., 43(2) (2019), 1036-1046.