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Keywords:
multiplicative lattice; Prüfer lattice; Prüfer integral domain
multiplicative lattice; Prüfer lattice; Prüfer integral domain
Summary:
We study the multiplicative lattices $L$ which satisfy the condition $ a=(a :\nobreak (a: \nobreak b))(a:b) $ for all $a,b\in L$. 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 $\mathbb {Z}$ or $\mathbb {R}$. A sharp lattice $L$ localized at its maximal elements are totally ordered sharp lattices. The converse is true if $L$ has finite character.
References:
[1] Ahmad, Z., Dumitrescu, T., Epure, M.: A Schreier domain type condition. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 55 (2012), 241-247. MR 2987061 | Zbl 1289.13001
[2] Anderson, D. D.: Abstract commutative ideal theory without chain condition. Algebra Univers. 6 (1976), 131-145. DOI 10.1007/BF02485825 | MR 0419310 | Zbl 0355.06022
[3] Anderson, D. D., Jayaram, C.: Principal element lattices. Czech. Math. J. 46 (1996), 99-109. DOI 10.21136/CMJ.1996.127274 | MR 1371692 | Zbl 0898.06008
[4] Dumitrescu, T.: A Bazzoni-type theorem for multiplicative lattices. Advances in Rings, Modules and Factorizations Springer Proceedings in Mathematics & Statistics 321. Springer, Cham (2020). DOI 10.1007/978-3-030-43416-8_6 | Zbl 07242233
[5] Engler, A. J., Prestel, A.: Valued Fields. Springer Monographs in Mathematics. Springer, Berlin (2005). DOI 10.1007/3-540-30035-X | MR 2183496 | Zbl 1128.12009
[6] Gilmer, R.: Multiplicative Ideal Theory. Pure and Applied Mathematics 12. Marcel Dekker, New York (1972). MR 0427289 | Zbl 0248.13001
[7] Halter-Koch, F.: Ideal Systems: An Introduction to Multiplicative Ideal Theory. Pure and Applied Mathematics, Marcel Dekker 211. Marcel Dekker, New York (1998). DOI 10.1515/9783110801194 | MR 1828371 | Zbl 0953.13001
[8] Jung, C. Y., Khalid, W., Nazeer, W., Tariq, T., Kang, S. M.: On an extension of sharp domains. Int. J. Pure Appl. Math. 115 (2017), 353-360. DOI 10.12732/ijpam.v115i2.12
[9] Larsen, M. D., McCarthy, P. J.: Multiplicative Theory of Ideals. Pure and Applied Mathematics 43. Academic Press, New York (1971). MR 0414528 | Zbl 0237.13002
[10] Olberding, B.: Globalizing local properties of Prüfer domains. J. Algebra 205 (1998), 480-504. DOI 10.1006/jabr.1997.7406 | MR 1632741 | Zbl 0928.13013
[11] Olberding, B., Reinhart, A.: Radical factorization in commutative rings, monoids and multiplicative lattices. Algebra Univers. 80 (2019), Article ID 24, 29 pages. DOI 10.1007/s00012-019-0597-1 | MR 3954364 | Zbl 1420.13008
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