Discrete valuation ring information

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an...

a discrete valuation ring. Conversely, the valuation ν : A → Z ∪ { ∞ } {\displaystyle \nu :A\rightarrow \mathbb {Z} \cup \{\infty \}} on a discrete valuation...

and a valuation ring has a discrete valuation group if and only if it is a discrete valuation ring. Very rarely, valuation ring may refer to a ring that...

terminology, a field K {\displaystyle K} with valuation v {\displaystyle v} is said to be Henselian if its valuation ring is Henselian. That is the case if and...

local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0. Any discrete valuation ring is a regular...

even a discrete valuation ring is not necessarily Japanese. Any quasi-excellent ring is a Nagata ring, so in particular almost all Noetherian rings that...

Discrete valuation Discrete valuation ring I-adic topology Weierstrass preparation theorem Noetherian ring Hilbert's basis theorem Artinian ring Ascending...

ring over k. Broadly speaking, regular local rings are somewhat similar to polynomial rings. Regular local rings are UFD's. Discrete valuation rings are...

local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. Equivalently, a local field...

In algebra, a Cohen ring is a field or a complete discrete valuation ring of mixed characteristic ( 0 , p ) {\displaystyle (0,p)} whose maximal ideal...

maximal ideal of A is principal. A is a discrete valuation ring (equivalently A is Dedekind.) A is a regular local ring. Let A be a noetherian integral domain...

nonzero ring in which every element is either a unit or nilpotent is a local ring. An important class of local rings are discrete valuation rings, which...

{\displaystyle R_{M}} is a Dedekind ring. But a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring (DVR), so the same local characterization...

of the discrete valuation ring R P i {\displaystyle R_{P_{i}}} and, being a quotient of a principal ring, is itself a principal ring. 6. Let k be...

a Cohen ring with the same residue field as the local ring. A Cohen ring is a field or a complete characteristic zero discrete valuation ring whose maximal...

x -1 ∈ O. Each such valuation ring is a discrete valuation ring and its maximal ideal is called a place of K/k. A discrete valuation of K/k is a surjective...

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its injective hull. The injective hull of the residue field of a discrete valuation ring ( R , m , k ) {\displaystyle (R,{\mathfrak {m}},k)} where m = x...

Then A {\displaystyle A} is a Krull ring if A p {\displaystyle A_{\mathfrak {p}}} is a discrete valuation ring for all p ∈ P {\displaystyle {\mathfrak...

series P and Q, f (P) ≤ f (Q) if and only if P divides Q. Any discrete valuation ring. Define f (x) to be the highest power of the maximal ideal M containing...

{\displaystyle K[[X]]} is a discrete valuation ring. The metric space ( R [ [ X ] ] , d ) {\displaystyle (R[[X]],d)} is complete. The ring R [ [ X ] ] {\displaystyle...

matters. (For a discrete valuation ring the topological space in question is the Sierpinski space of topologists. Other local rings have unique generic...

the original paper by Faltings, V was the integral closure of a discrete valuation ring in the algebraic closure of its quotient field, and m its maximal...

elements. Zp is an example of a complete discrete valuation ring of mixed characteristic. The integers, the ring of integers of any number field, and any...

example is the discrete two-point space. On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ring consists...