nLab ordered local integral domain

Definition

An ordered local integral domain is a local ring which is both an ordered local ring and a local integral domain: 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 if the product of two elements is equal to zero, than one of the two elements is equal to zero.

Properties

Unlike the theory of ordered fields, the theory of ordered local integral domains is a coherent theory.

See also

• ordered local ring

• local integral domain

• ordered field

• Archimedean ordered local integral domain