Given a commutative ring , is reduced or has a trivial nilradical if implies that for all .
For every natural number , implies that for all .
Let the function be defined as the ceiling of half of , . Then implies that , and for every natural number , the -th iteration of the function evaluated at is always equal to , , thus resulting in . Thus, the nilradical of is trivial.
As a result, the theory of a reduced ring is a coherent theory.
Every integral domain is a reduced ring. Thus, every field is a reduced ring.
An example of a reduced ring which is not an integral domain is the quotient ring .
Given a square-free integer , the integers modulo n is a reduced ring. Since for every integer , is a prefield ring, is an integral domain and thus a field if and only if is a prime number.
Given a discrete field , let denote its algebraic closure. Given a square-free polynomial , the quotient ring is a reduced ring. Since for every polynomial , is a prefield ring, is an integral domain and thus a field if and only if is a prime polynomial in , a monic polynomial of degree one; the resulting quotient ring is equivalent to .
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 August 19, 2024 at 14:56:00. See the history of this page for a list of all contributions to it.