Discrete valuation

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

satisfying the conditions

.

[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 νⁿ, 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

then the function defined as

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.

[edit] See also

• discrete valuation ring
• Valuation (mathematics)
• Valuation ring