nLab reduced local ring



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 010 \neq 1;

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

  • every nilpotent element is equal to zero; equivalently, every element which squares to zero is equal to zero.



The theory of reduced local rings is a coherent theory.

 See also


For a division algorithm for polynomials over reduced local rings:

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

Last revised on January 28, 2024 at 04:45:51. See the history of this page for a list of all contributions to it.