nLab Archimedean ordered reduced local ring

Idea

Since every ordered reduced local ring $R$ has characteristic zero, the positive integers $\mathbb{Z}_+$ are a subset of $R$, with injection $i:\mathbb{Z}_+ \hookrightarrow R$. An Archimedean ordered reduced local ring is an ordered reduced local ring which satisfies the archimedean property: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$.

An important class of Archimedean ordered reduced local rings are Archimedean ordered local integral domains, which are used in differential geometry and analysis to define analytic functions in the same way that Archimedean ordered Artinian local rings are used to define smooth functions. Archimedean ordered fields are the Archimedean ordered reduced local rings in which every non-invertible element is nilpotent, or equivalently, in which every non-invertible element is equal to zero.

See also

• Archimedean ordered field
• Archimedean ordered local ring
• Archimedean ordered local integral domain
• ordered reduced local ring
• formal power series