nLab ordered Artinian local ring

 Definition

An ordered Artinian local ring is a local ring which is both an ordered local ring and an Artinian local ring: a commutative ring $R$ with a strict weak order $\lt$ such that the positive elements form a multiplicative subset of $R$, the sum of two positive elements is positive, every element $a \in R$ is invertible if and only if it is positive or negative, and the set of non-invertible elements forms a nilradical.

See also

• ordered local ring

• Artinian local ring

• ordered field

• Archimedean ordered Artinian local ring