Discrete valuation

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In mathematics, a discrete valuation on an integral domain $A$ is a function

$\nu:A\to\mathbb Z\cup\{\infty\}$

satisfying the conditions

$\nu(x\cdot y)=\nu(x)+\nu(y)$

$\nu(x+y)\geq\mathrm{min}\big\{\nu(x),\nu(y)\big\}$

$\nu(x)=\infty\iff x=0$ .

[edit] Properties

Every discrete valuation ring gives rise to a discrete valuation, but not conversely. For example, if $K$ is a field, then the ring $K [[X, Y]]$ of power series over $K$ in two unknowns has a discrete valuation induced by the prime ideal $(X, Y)$ , and is even local, but is not a discrete valuation ring because it's not a principal ideal domain.

Consider a discrete valuation $ν$ on $A$ . If $B$ is the subset of all elements in $A$ with nonnegative valuation, then $B$ is also a subring of $A$ , and the set of all elements in $A$ with strictly positive valuation is a prime ideal of $B$ '.

[edit] Examples

If $A$ is the ring $Z$ of integers, then $ν n$ , defined as the largest value of $k$ such that $2 k$ divides $n$ , is a discrete valuation.
If $P$ is a prime ideal of $A$ satisfying the condition

$\bigcap_{n=1}^{\infty}P^{n} = 0 \in A$

then the function defined as

$\nu(x) = \max\{n: x \in P^{n}\}\quad x\in\mathbb{Z}$

is always finite for nonzero $x$ , an it can be proven to be a discrete valuation on $A$ . If $A$ is Noetherian, then every prime ideal of $A$ satisfies the above condition, so that every prime ideal induces a discrete valuation on $A$ .

Category: Commutative algebra

Discrete valuation

From Wikipedia, the free encyclopedia

[edit] Properties

[edit] Examples

[edit] See also

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