In
mathematics, a
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
of a
commutative ring is said to be irreducible if it cannot be written as the
intersection of two strictly larger ideals.
[.]
Examples
* Every
prime ideal is irreducible. Let
and
be ideals of a commutative ring
, with neither one contained in the other. Then there exist
and
, where neither is in
but the product is. This proves that a reducible ideal is not prime. A concrete example of this are the ideals
and
contained in
. The intersection is
, and
is not a prime ideal.
* Every irreducible ideal of a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
is a
primary ideal In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y'n'' is also an element of ''Q'', for some ''n'' > 0. Fo ...
,
and consequently for Noetherian rings an irreducible decomposition is a
primary decomposition
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many '' primary ideals'' (which are relate ...
.
* Every primary ideal of a
principal ideal domain is an irreducible ideal.
* Every irreducible ideal is
primal.
[. Theorem 1, p. 3.]
Properties
An element of an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
is
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
the ideal
generated by it is a non-zero prime ideal. This is not true for irreducible ideals; an irreducible ideal may be generated by an element that is not an
irreducible element
In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.
Relationship with prime elements
Irreducible elements should not be confus ...
, as is the case in
for the ideal
since it is not the intersection of two strictly greater ideals.
An ideal ''I'' of a ring ''R'' can be irreducible only if the
algebraic set
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
it defines is
irreducible (that is, any open subset is dense) for the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
, or equivalently if the closed space of
spec Spec may refer to:
*Specification (technical standard), an explicit set of requirements to be satisfied by a material, product, or service
**datasheet, or "spec sheet"
People
* Spec Harkness (1887-1952), American professional baseball pitcher
...
''R'' consisting of
prime ideals containing ''I'' is irreducible for the
spectral topology. The
converse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a definite or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical c ...
does not hold; for example the ideal of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s in two variables with vanishing terms of first and second order is not irreducible.
If k is an
algebraically closed field, choosing the
radical of an irreducible ideal of a
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
over k is exactly the same as choosing an
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is g ...
of the
affine variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
of its
Nullstelle in the affine space.
See also
*
Irreducible module
*
Irreducible space
*
Laskerian ring
References
Ring theory
Algebraic topology
{{Abstract-algebra-stub