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.

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.

If P is a prime ideal of A that satisfies the condition \bigcap_{n=1}^{\infty}P^{n} = 0, then \nu(x) = \max\{n: x \in P^{n}\} is always finite for nonzero x, so it's a discrete valuation on A. Conversely, if B is the set of all elements in A with nonnegative valuation, then B is a subring of A, and the set of all elements in A with strictly positive valuation is a prime ideal of B. 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.

Every discrete valuation ring gives rise to a discrete valuation, but not conversely. For example, if K is a field, then the ring of power series over K in two unknowns, K[[X, Y]], 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.

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