In mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...

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). Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, an algebraic set over a field K is the set of solutions in Kn (n-tuples of elements of K, of a set of simultaneous equations P1(X1, ...,Xn) = 0 P2(X1, ...,Xn) = 0 and so on up to Pm(X1, ...,Xn) = 0 for some integer m. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In mathematics, a projective space is a fundamental construction from any vector space. ... This article is about algebraic varieties. ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...

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. In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... In mathematics, the Lasker-Noether theorem provides a vast generalization of the fundamental theorem of arithmetic to embrace the rings of algebraic geometry. ... In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. ...