Discrete valuation
|
|
In mathematics, a discrete valuation on a commutative ring A is a function
- <math>\nu:A\to\mathbb Z\cup\{\infty\}<math>
satisfying the conditions
- <math>\nu(x\cdot y)=\nu(x)+\nu(y)<math>
- <math>\nu(x+y)\geq\mathrm{min}\big\{\nu(x),\nu(y)\big\}<math>
- <math>\nu(x)=\infty\iff x=0<math>.
For example, if A is the ring of integers, these properties are satisfied with ν(n) the largest value of k such that 2k divides n.
Every discrete valuation ring gives rise to a discrete valuation; but not conversely.pl:Waluacja dyskretna