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 in a commutative unital ring is termed a radical ideal if ... |
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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 and any positive integer , if , then . |
2 | equals its own radical | it equals its own radical in the whole ring | where denotes the radical of an ideal, i.e., the set of all elements for which some positive power lies inside the ideal. |
3 | quotient ring is reduced | the quotient ring by the ideal has trivial nilradical (that is, it is a reduced ring) | the quotient ring is a reduced ring: whenever and is a positive integer such that , then . |
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 of prime ideals indexed by a set such that . is allowed to be finite or infinite, and is also allowed to be empty. |
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 |
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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, and are radical ideals in but is not. 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, and are radical ideals in but is not. |
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 |
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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 |
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intersection-closed property of ideals in commutative unital rings | Yes | intersection of radical ideals is radical | Suppose is a (possibly finite, possibly infinite) collection of radical ideals in a commutative unital ring . Then the intersection is also a radical ideal in . |
contraction-closed property of ideals in commutative unital rings | Yes | Fill this in later | If is a homomorphism of commutative unital rings, and is a radical ideal in , then is a radical ideal in . |
intermediate subring condition for ideals | Yes | Fill this in later | Suppose is a radical ideal in a commutative unital ring and is a unital subring of that contains . Then, is also a radical ideal in . |
transfer condition for ideals | Yes | Fill this in later | If is a radical ideal in , and is a subring of , then is a radical ideal in . |