nLab reduced ring



Given a commutative ring RR, RR is reduced or has a trivial nilradical if xx=0x \cdot x = 0 implies that x=0x = 0 for all xRx \in R.


For every natural number nn, x n+1=0x^{n + 1} = 0 implies that x=0x = 0 for all xRx \in R.


Let the function f:f:\mathbb{N} \to \mathbb{N} be defined as the ceiling of half of nn, f(n)n/2f(n) \coloneqq \lceil n/2 \rceil. Then x n+1=0x^{n + 1} = 0 implies that x f(n+1)=0x^{f(n + 1)} = 0, and for every natural number nn, the (n+1)(n + 1)-th iteration of the function ff evaluated at n+1n + 1 is always equal to 11, f n+1(n+1)=1f^{n + 1}(n + 1) = 1, thus resulting in x f n+1(n+1)=x=0x^{f^{n + 1}(n + 1)} = x = 0. Thus, the nilradical of RR is trivial.

As a result, the theory of a reduced ring is a coherent theory.


See also

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


Last revised on July 31, 2023 at 10:45:32. See the history of this page for a list of all contributions to it.