Prime ideal
Definition
Equivalent definitions in tabular format
No. | Shorthand | An ideal in a commutative unital ring is termed a prime ideal if ... | An ideal ![]() ![]() |
---|---|---|---|
1 | product of two elements version | It is a proper ideal and whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2 | product of two ideals version | It is a proper ideal and whenever the product of two ideals is contained in it, one of the ideals is contained in it. | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | product of finitely many elements version | It is a proper ideal and whenever the product of finitely many elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | product of finitely many ideals version | It is a proper ideal and whenever the product of finitely many ideals is contained in it, one of the ideals is contained in it. | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | complement saturated subset version | It is a proper ideal and its complement is a multiplicatively closed saturated subset (that is, is closed with respect to the operations of multiplication and factorization). | ![]() ![]() ![]() ![]() ![]() ![]() |
6 | quotient is integral domain | The quotient ring is an integral domain, i.e., a nonzero ring where the product of any two nonzero elements is nonzero. | The quotient ring ![]() |
The set of prime ideals
Further information: Spectrum of a commutative unital ring
The set of prime ideals in a commutative unital ring is termed its spectrum. The spectrum is more than just a set, it has the structure of a topological space. In fact, it is a locally ringed space.
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: integral domain | View other quotient-determined properties of ideals in commutative unital rings
Relation with other properties
Stronger properties
Weaker properties
Metaproperties
Intersection-closedness
This property of ideals in commutative unital rings is not closed under taking arbitrary intersections; in other words, an arbitrary intersection of ideals with this property need not have this property
An intersection of prime ideals need not be prime. In fact, an ideal is an intersection of prime ideals iff it is a radical ideal.
Contraction-closedness
This property of ideals in commutative unital rings is contraction-closed: a contraction of an ideal with this property, also has this property
View other contraction-closed properties of ideals in commutative unital rings
If is a homomorphism of commutative unital rings, and
is a prime ideal of
, then the contraction of
to
, denoted
, (same as
) is a prime ideal in
.
For full proof, refer: primeness is contraction-closed
Note that this implies the intermediate subring condition and the transfer condition on ideals.
Intermediate subring condition
This property of ideals satisfies the intermediate subring condition for ideals: if an ideal has this property in the whole ring, it also has this property in any intermediate subring
View other properties of ideals satisfying the intermediate subring condition
If an ideal is prime in the whole ring, it is also prime in any intermediate subring. This is related to the fact that any subring of an integral domain is an integral domain.
Transfer condition
This property of ideals satisfies the transfer condition for ideals: if an ideal satisfies the property in the ring, its intersection with any subring satisfies the property inside that subring
If is a prime ideal in
, and
is any subring of
, then
is a prime ideal in
. Note that this implies the intermediate ring condition as well.
Effect of property operators
The intersection-closure
Applying the intersection-closure to this property gives: radical ideal
An ideal in a commutative unital ring is expressible as an intersection of prime ideals iff it is a radical ideal.
Further information: Intersection of prime equals radical
In particular kinds of rings
In rings of integers
Further information: Prime ideal in ring of integers
A ring of integers in a number field is a Dedekind domain, hence any nonzero prime ideal in this ring is a maximal ideal.
In affine rings
Further information: Prime ideal in affine ring
Ring property assumed | Nature of prime ideals |
---|---|
Zero-dimensional ring | maximal |
One-dimensional domain | Zero or maximal |
Principal ideal domain | Zero or maximal |
Dedekind domain | Zero or maximal |
References
- Book:Eisenbud, Page 12
External links
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Definition links
- Basic definitions in commutative algebra
- Standard terminology
- Properties of ideals in commutative unital rings
- Quotient-determined properties of ideals in commutative unital rings
- Contraction-closed properties of ideals in commutative unital rings
- Properties of ideals satisfying intermediate subring condition
- Properties of ideals satisfying transfer condition