Radical ideal
From Commalg
Definition
Equivalent definitions in tabular format
No. | Shorthand | An ideal in a commutative unital ring is termed a radical ideal if ... | An ideal ![]() ![]() |
---|---|---|---|
1 | closed under taking roots | whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal | For any ![]() ![]() ![]() ![]() |
2 | equals its own radical | it equals its own radical in the whole ring | ![]() ![]() |
3 | quotient ring is reduced | the quotient ring by the ideal has trivial nilradical (that is, it is a reduced ring) | the quotient ring ![]() ![]() ![]() ![]() ![]() |
4 | intersection of prime ideals | it can be expressed as an intersection of prime ideals. The intersection is allowed to be finite or infinite. An empty intersection, which would give the whole ring, is also allowed. | There exists a collection ![]() ![]() ![]() ![]() |
Note that unlike for the definition of prime ideals, we do not require a radical ideal to be proper, so the whole ring is always a radical ideal in itself.
This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra
This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings
This property of an ideal in a ring is equivalent to the property of the quotient ring being a/an: reduced ring | View other quotient-determined properties of ideals in commutative unital rings
Examples
Important ring types
Property of commutative unital rings | Characterization of radical ideals in such a ring | Examples |
---|---|---|
principal ideal domain (i.e., an integral domain in which every ideal is a principal ideal) | either the zero ideal or a principal ideal generated by a square-free element, i.e., the generating element should be a product of distinct primes | In the ring of integers, the radical ideals are precisely the zero ideal and the ideals generated by square-free numbers. Thus, for instance, ![]() ![]() ![]() ![]() In the (univariate) polynomial ring over a field, the radical ideals are precisely the zero ideal and the ideals generated by square-free polynomials, which can be described as polynomials with no repeated roots over the algebraic closure, i.e., polynomials that are relatively prime to their derivative polynomial. Thus, for instance, ![]() ![]() ![]() ![]() |
Dedekind domain (roughly speaking, admits unique factorization of ideals into prime ideals) | either zero or a product of distinct prime ideals | Fill this in later |
unique factorization domain (unique factorization of elements into primes) | principal ideals generated by square-free elements are some, but not all, of the radical ideals. | In the multivariate polynomial ring over a field, ... |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions | Collapse |
---|---|---|---|---|---|
Maximal ideal | proper ideal not contained in any bigger proper ideal quotient is a field |
(via prime ideal) | (via prime ideal) | ? | |
Prime ideal | quotient is an integral domain | prime implies radical | radical not implies prime | ? | |
Intersection of maximal ideals | intersection of maximal ideals | Jacobson ring |
Incomparable properties
Metaproperties
Metaproperty | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
intersection-closed property of ideals in commutative unital rings | Yes | intersection of radical ideals is radical | Suppose ![]() ![]() ![]() ![]() |
contraction-closed property of ideals in commutative unital rings | Yes | Fill this in later | If ![]() ![]() ![]() ![]() ![]() |
intermediate subring condition for ideals | Yes | Fill this in later | Suppose ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
transfer condition for ideals | Yes | Fill this in later | If ![]() ![]() ![]() ![]() ![]() ![]() |