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.

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). Irreducible variety is therefore a pleonasm.

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.