# Some applications of the ultrafilter topology on spaces of valuation domains, Part II

Carmelo Antonio Finocchiaro^{[1]}; Marco Fontana^{[2]}

- [1] C.A.F. - Dipartimento di Matematica Università degli studi Roma Tre Largo San Leonardo Murialdo 1, 00146 Roma, Italy
- [2] M.F. - Dipartimento di Matematica Università degli studi Roma Tre Largo San Leonardo Murialdo 1, 00146 Roma, Italy

Actes des rencontres du CIRM (2010)

- Volume: 2, Issue: 2, page 111-114
- ISSN: 2105-0597

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topFinocchiaro, Carmelo Antonio, and Fontana, Marco. "Some applications of the ultrafilter topology on spaces of valuation domains, Part II." Actes des rencontres du CIRM 2.2 (2010): 111-114. <http://eudml.org/doc/196293>.

@article{Finocchiaro2010,

abstract = {Let $K$ be a field and $A$ be a subring of $K$. In the present note, we present the main applications of the so called ultrafilter topology on the space $\{\rm Zar\}(K|A)$, introduced in the previous Part I. After recalling that $\{\rm Zar\}(K|A)$ is a spectral space, we give an explicit description of $\{\rm Zar\}(K|A)$ as the prime spectrum of a ring (even in the case when the quotient field of $A$ is a proper subfield of $K$). Moreover, we provide applications of the topological material previously introduced to the study of representations of integrally closed domains and valuative semistar operations.},

affiliation = {C.A.F. - Dipartimento di Matematica Università degli studi Roma Tre Largo San Leonardo Murialdo 1, 00146 Roma, Italy; M.F. - Dipartimento di Matematica Università degli studi Roma Tre Largo San Leonardo Murialdo 1, 00146 Roma, Italy},

author = {Finocchiaro, Carmelo Antonio, Fontana, Marco},

journal = {Actes des rencontres du CIRM},

keywords = {Valuation domain; (semi)star operation; prime spectrum; Zariski topology; constructible topology; filter and ultrafilter; Prüfer domain},

language = {eng},

number = {2},

pages = {111-114},

publisher = {CIRM},

title = {Some applications of the ultrafilter topology on spaces of valuation domains, Part II},

url = {http://eudml.org/doc/196293},

volume = {2},

year = {2010},

}

TY - JOUR

AU - Finocchiaro, Carmelo Antonio

AU - Fontana, Marco

TI - Some applications of the ultrafilter topology on spaces of valuation domains, Part II

JO - Actes des rencontres du CIRM

PY - 2010

PB - CIRM

VL - 2

IS - 2

SP - 111

EP - 114

AB - Let $K$ be a field and $A$ be a subring of $K$. In the present note, we present the main applications of the so called ultrafilter topology on the space ${\rm Zar}(K|A)$, introduced in the previous Part I. After recalling that ${\rm Zar}(K|A)$ is a spectral space, we give an explicit description of ${\rm Zar}(K|A)$ as the prime spectrum of a ring (even in the case when the quotient field of $A$ is a proper subfield of $K$). Moreover, we provide applications of the topological material previously introduced to the study of representations of integrally closed domains and valuative semistar operations.

LA - eng

KW - Valuation domain; (semi)star operation; prime spectrum; Zariski topology; constructible topology; filter and ultrafilter; Prüfer domain

UR - http://eudml.org/doc/196293

ER -

## References

top- A. Fabbri, Integral domains with a unique Kronecker function ring, J. Pure Appl. Algebra215 (2011), 1069-1084. Zbl1211.13002MR2747239
- C. A. Finocchiaro, M. Fontana, Some applications of the ultrafilter topology on spaces of valuation domains, Part I, this volume.
- C. A. Finocchiaro, M. Fontana, K. A. Loper, Ultrafilter and constructible topologies on spaces of valuation domains, submitted. Zbl1310.13006
- M. Fontana, K. A. Loper, Cancellation properties in ideal systems: a classification of e.a.b. semistar operations, J. Pure Appl. Algebra213 (2009), no. 11, 2095–2103. Zbl1187.13003MR2533308
- F. Halter–Koch, Kronecker function rings and generalized integral closures, Comm. Algebra31 (2003), 45-49. Zbl1073.13507MR1969212
- Melvin Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc.142 (1969), 43–60. Zbl0184.29401MR251026

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