symmetric monoidal (∞,1)-category of spectra
A local integral domain is a commutative ring which is both a local ring and a integral domain, hence a commutative ring such that:
The ring is nontrivial $0 \neq 1$;
if the sum of two elements $x + y$ is equal to $1$, then $x$ is invertible or $y$ is invertible;
If the product of two elements $x \cdot y$ is equal to $0$, then $x = 0$ or $y = 0$.
Every field is a local integral domain which is also a Artinian ring.
Given a field $K$, the ring of power series $K[[x]]$ is a local integral domain which is not a field.
In general, given a local integral domain $R$, the ring of power series $R[[x]]$ is a local integral domain.
Every discrete valuation ring is a local integral domain with a Dedekind-Hasse norm.
Every finite local integral domain is a finite field.
Unlike the theory of Heyting fields, the theory of local integral domains is a coherent theory. It is the condition that a local integral domain is an Artinian ring that fails in coherent logic, since it relies on either negation to refer to non-invertible elements (see Artinian local ring), proper ideals/maximal ideals, or the natural numbers to refer to the descending chain condition. Even if one has the natural numbers around to define a dependent sequence of monomorphisms for the descending chain condition, such as working internally in an arithmetic pretopos, it is not necessarily the case that every such Artinian local ring has a maximal ideal or that Nakayama's lemma holds in coherent logic.
commutative ring | reduced ring | integral domain |
---|---|---|
local ring | reduced local ring | local integral domain |
Artinian ring | semisimple ring | field |
Weil ring | field | field |
Last revised on January 12, 2023 at 17:46:58. See the history of this page for a list of all contributions to it.