Irreducible component
From Wikipedia, the free encyclopedia
In mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation
- XY = 0
is the union of the two lines
- X = 0
and
- Y = 0.
[edit] Definition of irreducibility for topological spaces
A subset F of a topological space X is reducible if it can be written as a union <math>F = F_1 \cup F_2</math> of two closed proper subsets <math>F_1</math>, <math>F_2</math> of <math>F</math> (closed in the subspace topology). That is, <math>F</math> is reducible if it can be written as a union <math>F = (G_1\cap F)\cup(G_2\cap F)</math> where <math>G_1,G_2</math> are closed subsets of <math>X</math>, neither of which contains <math>F</math>.
A subset of a topological space is irreducible (or hyperconnected) if it is not reducible.
An irreducible component of a topological space is a maximal irreducible subset. If a subset is irreducible, its closure is, so irreducible components are closed.
As an example, consider <math>\{ (x,y)\in\mathbb{R}^2 : xy = 0 \}</math> with the subspace topology. This space is a union of two lines <math>\{ (x,y)\in\mathbb{R}^2 : x = 0 \}</math> and <math>\{ (x,y)\in\mathbb{R}^2 : y = 0 \}</math>, which are proper closed subsets. So this space is reducible, and thus not irreducible.
[edit] Use in Algebraic Geometry
In algebraic geometry, any algebraic set, in affine space or projective space, is the union of a finite number of irreducible components, which are algebraic varieties in the strict sense of being irreducible (in the affine case, this is the same as the condition that their coordinate rings are integral domains).
As a matter of commutative algebra, the primary decomposition of an ideal gives rise to the decomposition into irreducible components; and is somewhat finer in the information it gives, since it is not limited to radical ideals.
This article incorporates material from irreducible on PlanetMath, which is licensed under the GFDL.
This article incorporates material from Irreducible component on PlanetMath, which is licensed under the GFDL.
