symmetric monoidal (∞,1)-category of spectra
A reduced local ring is a commutative ring which is both a local ring and a reduced ring, 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;
every nilpotent element is equal to zero; equivalently, every element which squares to zero is equal to zero.
Every Heyting field is a reduced local ring which is also a Artinian ring.
Given a discrete field $K$, the ring of power series $K[[x]]$ is a reduced local ring which is not a Heyting field.
The theory of reduced local rings is a coherent theory.
For a division algorithm for polynomials over reduced local rings:
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 28, 2024 at 04:45:51. See the history of this page for a list of all contributions to it.