nLab ordered reduced local ring

Definition

An ordered reduced local ring is a local ring which is both an ordered local ring and an reduced 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 every nilpotent element is equal to zero.

Properties

Unlike the theory of ordered fields, the theory of ordered reduced local rings is a coherent theory.

See also

• ordered local ring

• reduced local ring

• ordered field

• Archimedean ordered reduced local ring