nLab local integral domain

Definition

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 $y = 0$.

Examples

• 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.

Properties

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.

See also

• ordered local integral domain

• discrete valuation ring

commutative ring	reduced ring	integral domain
local ring	reduced local ring	local integral domain
Artinian ring	semisimple ring	field
Weil ring	field	field