Discrete valuation ring

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

This means a DVR is an integral domain

R

{\displaystyle R}

that satisfies any and all of the following equivalent conditions:

R

{\displaystyle R}

is a local ring, a principal ideal domain, and not a field.

R

{\displaystyle R}

is a valuation ring with a value group isomorphic to the integers under addition.

R

{\displaystyle R}

is a local ring, a Dedekind domain, and not a field.

R

{\displaystyle R}

is Noetherian and a local domain whose unique maximal ideal is principal, and not a field.

R

{\displaystyle R}

is integrally closed, Noetherian, and a local ring with Krull dimension one.

R

{\displaystyle R}

is a principal ideal domain with a unique non-zero prime ideal.

R

{\displaystyle R}

is a principal ideal domain with a unique irreducible element (up to multiplication by units).

R

{\displaystyle R}

is a unique factorization domain with a unique irreducible element (up to multiplication by units).

R

{\displaystyle R}

is Noetherian, not a field, and every nonzero fractional ideal of R is irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it.