Reduced ring
From Commalg
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 is said to be reduced if it satisfies the following condition:
for any
Relation with other properties
Stronger properties
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.