- Irreducible component
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.

The notion of irreducibility is stronger than connectedness.

**Definition**A topological space "X" is

**reducible**if it can be written as a union $F\; =\; F\_1\; cup\; F\_2$ of two closed proper subsets $F\_1$, $F\_2$ of $F$.A topological space is**irreducible**(or**hyperconnected**) if it is not reducible. Equivalently, all open subsets of "X" are dense.A subset "F" of a topological space "X" is called irreducible or reducible, if "F" considered as a topological space via the subspace topology has the corresponding property in the above sense. That is, $F$ is reducible if it can be written as a union $F\; =\; (G\_1cap\; F)cup(G\_2cap\; F)$ where $G\_1,G\_2$ are closed subsets of $X$, neither of which contains $F$.

An

**irreducible component**of atopological space is amaximal irreducible subset. If a subset is irreducible, its closure is, so irreducible components are closed.**Use in algebraic geometry**In general

algebraic variety orscheme "X" is the union of its irreducible components "X_{i}". In most cases occurring in "practice", namely for allnoetherian scheme s, there are finitely many irreducible components. There is the following description of irreducible affine varieties or schemes "X = Spec A": "X" is irreducibleiff thecoordinate ring "A" of "X" has one minimalprime ideal . This follows from the definition of theZariski topology . In particular, if "A" has nozero divisor s, "Spec A" is irreducible, because then the zero-ideal is the minimal prime ideal.As a matter of

commutative algebra , theprimary 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 toradical ideal s.An affine variety or scheme "X = Spec A" is connected iff "A" has no nontrivial (i.e. ≠0 or 1)

idempotent s. Geometrically, a nontrivial idempotent "e" corresponds to the function on "X" which is equal to "1" on some connected component(s) and "0" on others.Irreducible components serve to define the dimension of schemes.

**Examples**The irreducibility depends much on actual topology on some set. For example, possibly contradicting the intuition, the real numbers (with their usual topology) are reducible: for example the open interval ("-1, 1") is not dense, its closure is the closed interval ["-1, 1"] .

However, the notion is fundamental and more meaningful in

algebraic geometry : consider the variety:"X" := {"x · y = 0"}(a subset of the affine plane, "x" and "y" are the variables) endowed with the "Zariski topology ". It is reducible, its irreducible components are its closed subset {"x = 0"} and {"y = 0"}.This can also be read off the coordinate ring "k" ["x,y"] /("xy") (if the variety is defined over a field "k"), whose minimal prime ideals are ("x") and ("y").

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