BIGpedia - Discrete valuation - Encyclopedia and Dictionary Online

In mathematics, a discrete valuation on a commutative ring A is a function

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

satisfying the conditions

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

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

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

For example, if A is the ring of integers, these properties are satisfied with ν(n) the largest value of k such that 2^k divides n.

Every discrete valuation ring gives rise to a discrete valuation; but not conversely.

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01-04-2007 01:21:04