Abstract
We study the multiplicative lattices L which satisfy the condition a = (a : (a : b))(a : b) for all a, b ∈ L. Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group ℤ or ℝ. A sharp lattice L localized at its maximal elements are totally ordered sharp lattices. The converse is true if L has finite character.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Z. Ahmad, T. Dumitrescu, M. Epure: A Schreier domain type condition. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 55 (2012), 241–247.
D. D. Anderson: Abstract commutative ideal theory without chain condition. Algebra Univers. 6 (1976), 131–145.
D. D. Anderson, C. Jayaram: Principal element lattices. Czech. Math. J. 46 (1996), 99–109.
T. Dumitrescu: A Bazzoni-type theorem for multiplicative lattices. Advances in Rings, Modules and Factorizations. Springer Proceedings in Mathematics & Statistics 321. Springer, Cham, 2020.
A. J. Engler, A. Prestel: Valued Fields. Springer Monographs in Mathematics. Springer, Berlin, 2005.
R. Gilmer: Multiplicative Ideal Theory. Pure and Applied Mathematics 12. Marcel Dekker, New York, 1972.
F. Halter-Koch: Ideal Systems: An Introduction to Multiplicative Ideal Theory. Pure and Applied Mathematics, Marcel Dekker 211. Marcel Dekker, New York, 1998.
C. Y. Jung, W. Khalid, W. Nazeer, T. Tariq, S. M. Kang: On an extension of sharp domains. Int. J. Pure Appl. Math. 115 (2017), 353–360.
M. D. Larsen, P. J. McCarthy: Multiplicative Theory of Ideals. Pure and Applied Mathematics 43. Academic Press, New York, 1971.
B. Olberding: Globalizing local properties of Prüfer domains. J. Algebra 205 (1998), 480–504.
B. Olberding, A. Reinhart: Radical factorization in commutative rings, monoids and multiplicative lattices. Algebra Univers. 80 (2019), Article ID 24, 29 pages.
Acknowledgement
We thank the referee whose suggestions improved the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dumitrescu, T., Epure, M. A Class of Multiplicative Lattices. Czech Math J 71, 591–601 (2021). https://doi.org/10.21136/CMJ.2021.0034-20
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2021.0034-20