Abstract
In 1994 A. Okabe and R. Matsuda [22] introduced the notion of semistar operation; see also, [21] and [20]. This concept extends the classical concept of star operation, as developed in Gilmer’s book [12], and hence the related classical theory of ideal systems based on the works of W. Krull, E. Noether, H. Prüfer, and P. Lorenzen from the 1930’s. For a systematic treatment of these ideas, see the books by P. Jaffard [17] and F. Halter-Koch [14], where a complete and updated bibliography is available.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988), 2535–2553.
D. D. Anderson and D. F. Anderson, Some remarks on star operations and the class group, J. Pure Appl. Algebra 51 (1988), 27–33.
D. D. Anderson and S. J. Cook, Star operations and their induced lattices, preprint.
N. Bourbaki, Algèbre Commutative, Hermann, Paris, 1961–1965.
V. Barucci, D. Dobbs, and M. Fontana, Conducive integral domains as pullbacks, Manuscripta Math. 54 (1986), 261–277.
E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form “D + M”, Michigan Math. J. 20 (1973), 79–95.
Wang Fanggui and R.L. McCasland, On w-modules over strong Mori domains, Comm. Algebra 25 (1997), 1285–1306.
M. Fontana, Kaplansky ideal transform: a survey, M. Dekker Lect. Notes 205 (1999), 271–306.
M. Fontana, J. Huckaba, and I. Papick, Prüfer Domains, M. Dekker, New York, 1997.
S. Gabelli, Prüfer (##) domains and localizing systems of ideals, M. Dekker Lect. Notes, 205 (1999), 391–410.
J. Garcia, P. Jara, and E. Santos, Prüfer * -multiplication domains and torsion theories, Comm. Algebra, 27 (1999), 1275–1295.
R. Gilmer, Multiplicative Ideal Theory, M. Dekker, New York, 1972.
F. Halter-Koch, Kronecker function rings and generalized integral closures, preprint 1999.
F. Halter-Koch, Ideal Systems: An Introduction to Multiplicative Ideal Theory, M. Dekker, New York, 1998.
W. Heinzer and J. Ohm, An essential ring which is not a v-multiplication ring, Can. J. Math., 25 (1973), 856–861.
J.M. Garcia Hernandez, Radicales de anillos y modulos noetherianos relativos, Ph.D. Thesis, Univ. Granada (1995).
P. Jaffard, Les Systèmes d’Idéaux, Dunod, Paris, 1960.
B. Kang, *-Operations On Integral Domains, Ph.D. Dissertation, The University of Iowa, 1987.
R. Matsuda, Kronecker function rings of semistar operations on rings, Algebra Colloquium, 5 (1998), 241–254.
R. Matsuda and I. Sato, Note on star-operations and semistar operations, Bull. Fac. Sci. Ibaraki Univ., 28 (1996), 155–161.
R. Matsuda and T. Sugatani, Semistar operations on integral domains, II. Math. J. Toyama Univ., 18 (1995), 155–161.
A. Okabe and R. Matsuda, Semistar-operations on integral domains, Math. J. Toyama Univ. 17 (1994), 1–21.
A. Okabe and R. Matsuda, Kronecker function rings of semistar operations, Tsukuba J. Math., 21 (1997), 529–548.
N. Popescu, A characterization of generalized Dedekind domains, Rev. Roumaine Math. Pures Appl. 29 (1984), 777–786.
B. Stenström, Rings of Quotients, Springer, Berlin 1975.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Fontana, M., Huckaba, J.A. (2000). Localizing Systems and Semistar Operations. In: Chapman, S.T., Glaz, S. (eds) Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications, vol 520. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3180-4_8
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3180-4_8
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4835-9
Online ISBN: 978-1-4757-3180-4
eBook Packages: Springer Book Archive