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Everything about Valuation ring
Concept in algebra
In abstract algebra, a valuation ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x−1 belongs to D. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Valuation ringA 5-minute orientation
- Skim a recent paper: Algebraic valuation ring extensions as limits of complete intersection algebrasInfluential work since 2022
- Pick up a classic book: Valuation theoryThe most-reprinted book on the subject
- Read the landmark paper: Vanishing Cycles and Geometry of Curves Over a Discrete Valuation RingCited 35 times
- Go sideways: Integrally closed domainA closely related topic
- Go sideways: Integral elementA closely related topic
📖 Definitions
Noun An integral domain D such that for every element x of its field of fractions F, at least one of x or x belongs to D.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
Vanishing Cycles and Geometry of Curves Over a Discrete Valuation Ring
Modules over a complete discrete valuation ring
Global dimension of tiled orders over a discrete valuation ring
Dual modules over a valuation ring. I
Finitely generated rings over a valuation ring
Global dimension of triangular orders over a discrete valuation ring
An invariant for modules over a discrete valuation ring
ON GROUP REPRESENTATIONS OVER A VALUATION RING
🆕 Influential since 2022
Algebraic valuation ring extensions as limits of complete intersection algebras
Linear Differential Equation with Inhomogeneity in the Form of a Formal Power Series Over a Ring with Non-Archimedean Valuation
Exploring barriers to valuation technology adoption in property valuation: insights from practitioners
Graded almost valuation rings
On some valuation-like rings in certain ring extensions
📚 Books worth reading
Valuation theoryOtto Endler · 1972- 📕Modules over valuation ringsLászló Fuchs · 1983
Manis valuations and Prüfer extensionsManfred Knebusch, Digen Zhang · 2002
Almost ring theoryOfer Gabber, Lorenzo Ramero · 2003
Non-commutative valuation rings and semi-hereditary ordersHidetoshi Marubayashi · 1997
Places and valuations in noncommutative ring theoryJan Van Geel · 1981- 📕Modules quadratiques sur un anneau de valuation discrèteW. Faivre · 1975
- 📕Hereditarily-pythagorean fields and orderings of higher levelEberhard Becker · 1978
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
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