Abstract
The all over idea of the present chapter is to associate to any ring extension A ⊂ R a commuting square of ring extensions 
such that B is Prüfer in T and there exists a process v↦v
∗ which associates with v in a suitable family \(\mathfrak{M}\) of valuations of R over A a special valuation v of T over B such that v
∗∘ j = v. (“Over A” means that A
v
⊃ A.)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
“Semistar operation” here has another meaning than in the classical literature, cf. Comments, p. 139.
- 2.
In [Vol. I, Chap. III §3] we only aimed at giving some interesting examples of BM-valuations, and thus have been more brief than now.
- 3.
v(I) had been defined in Sect. 1.
- 4.
We call any valuation equivalent to v | H for some convex subgroup H of Γ with H ⊃ c v (Γ) a primary specialization of v (cf. [HK] for this terminology).
- 5.
X∕1 denotes the image of X in \(R(X,\mathcal{G})\) under the map \(j_{\mathcal{G}}^{{^\prime}}\).
- 6.
Recall that T ∗ denotes the group of units of T.
- 7.
In most of the literature I δ is denoted by I v or I v (the “v-operation”). We refuse to use the letter v here, since all too often in our book v denotes a valuation.
- 8.
In most of the literature an additive notation is chosen and the order relation is reversed: 0 = A is the neutral element; I ≤ J iff I ⊃ J.
- 9.
Below (Definition 6) we will introduce a subgroup of \(\tilde{D}_{{\ast}}(A,R)\) conisting of star invertible “fractional ideals”, for which we reserve the notation D ∗(A, R).
- 10.
Notice that L is not an ideal of A in the usual sense.
- 11.
In much of the literature the word “complete” is used instead of “convex”. We are afraid of conflict with the topological meaning of “complete”. (Any family Φ of valuations on R gives a uniform topology on R.)
- 12.
Recall that for H ∈ J(B,T) we use R ∩ H as an abbreviation of \({\varphi }^{-1}(H)\) , and that IB means \(\varphi (I)B\).
- 13.
A notable exception is the axiom St4′ which may loose its sense, since in general \(\mathcal{G}\) may not contain enough principal ideals.
References
D. Anderson, M. Fontana, M. Zafrullah, Some remarks on Prüfer ∗-multiplication domains and class groups. J. Algebra 319, 272–295 (2008)
N. Bourbaki, Algebrè Commutative, Chap. 1–7 (Hermann, Paris, 1961–1965)
M. Fontana, K.A. Loper, Kronecker function rings: a general approach, in Ideal Theoretic Methods in Commutative Algebra, ed. by D.D. Anderson, I.J. Papick. Lecture Notes in Pure and Applied Mathematics, vol. 220 (Marcel Decker, New York, 2001), pp. 189–205
S. Gabelli, G. Picozza, Star stable domains. J. Pure Appl. Algebra 208, 853–866 (2007)
R. Gilmer, Multiplicative Ideal Theory (Marcel Dekker, New York, 1972)
F. Halter-Koch, Kronecker function rings and generalized integral closures. Commun. Algebra 31, 45–59 (2003)
F. Halter-Koch, Ideal Systems; An Introduction to Multiplicative Ideal Theory (Marcel Dekker, New York, 1996)
R. Huber, M. Knebusch, On valuation spectra. Contemp. Math. 155, 167–206 (1994)
J.A. Huckaba, Commutative Rings with Zero Divisors (Marcel Dekker, New York, 1988)
L. Kronecker, Grundzüge einer arithmetischen Theorie der algebraischen Größen. J. Reine Angew. Math. 92, 1–122 (1988)
F. Mertens, Über einen algebraischen Satz. S.-B. Akad. Wiss. Wien (2a) 101, 1560–1566 (1892)
M. Nagata, Local Rings (Interscience Publ./Wiley, New York, 1962)
B. van der Waerden, Algebra, Zweiter Teil (Springer, Berlin, 1959)
H. Weyl, Algebraic Theory of Numbers (Princeton University Press, Princeton, 1940)
M. Zafrullah, t-invertibility and Bazzoni-like statements. J. Pure Appl. Algebra 214, 654–657 (2010)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Knebusch, M., Kaiser, T. (2014). Kronecker Extensions and Star Operations. In: Manis Valuations and Prüfer Extensions II. Lecture Notes in Mathematics, vol 2103. Springer, Cham. https://doi.org/10.1007/978-3-319-03212-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-03212-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03211-5
Online ISBN: 978-3-319-03212-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)