nLab local integral domain



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 010 \neq 1;

  • if the sum of two elements x+yx + y is equal to 11, then xx is invertible or yy is invertible;

  • If the product of two elements xyx \cdot y is equal to 00, then x=0x = 0 or y=0y = 0.



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.

While every ring homomorphism between fields is an injection, it is not the case that every ring homomorphism between local integral domains is an injection. The ring homomorphism which takes a local integral domain to its residue field is an injection if and only if the local integral domain is already a field.

See also

commutative ringreduced ringintegral domain
local ringreduced local ringlocal integral domain
Artinian ringsemisimple ringfield
Weil ringfieldfield

Last revised on February 1, 2024 at 15:04:19. See the history of this page for a list of all contributions to it.