Reduced ring

This article is about a standard (though not very rudimentary) definition in commutative algebra. The article text may, however, contain more than just the basic definition
View a complete list of semi-basic definitions on this wiki

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

The property of being an ideal for which the quotient ring has this property is: radical ideal

Definition

Symbol-free definition

A commutative unital ring is said to be reduced if it satisfies the following equivalent conditions:

• The nilradical of the ring is the zero ideal
• There are no nilpotents other than the zero element
• The zero ideal is a radical ideal

Definition with symbols

A commutative unital ring $R$ is said to be reduced if it satisfies the following condition:

$x^n = 0 \implies x = 0$ for any $x \in R$

Relation with other properties

Stronger properties

• Integral domain
• Semisimple ring

Metaproperties

Closure under taking subrings

Any subring of a commutative unital ring with this property, also has this property
View other subring-closed properties of commutative unital rings

Any subring of a reduced ring is reduced. That's because an element in the subring that is nilpotent, is also nilpotent in the whole ring.