Irreducibility and Noetherian decomposition

A nonempty closed set X is irreducible if it is not the union of two proper closed subsets. Equivalently, any two nonempty open subsets of X meet. Consequently each nonempty open subset is dense: otherwise its closure and its complement would express X as a union of two proper closed sets. This is a Zariski-topological property, not a statement that a real drawing has only one visible arc. A polynomial irreducible over the reals may also factor over the algebraic closure.

If a product fg vanishes on X, then X lies in the union of the zero loci of f and g. Irreducibility forces one factor to vanish on all of X, proving that I(X) is prime. Conversely, a decomposition into two proper closed subsets provides a polynomial vanishing on each piece but not everywhere; their product belongs to I(X), violating primality. Thus irreducibility corresponds exactly to a domain coordinate ring. Reduced rings alone allow zero divisors and can therefore describe unions.

Affine algebraic sets are Noetherian topological spaces: a descending sequence of closed subsets eventually stabilizes. Their vanishing ideals form an ascending sequence, and the polynomial ring is Noetherian by Hilbert's basis theorem. If a closed set could not be expressed as a finite union of irreducibles, split it into two smaller closed sets. At least one would still be a counterexample, allowing an infinite strict descending chain. The contradiction gives a finite decomposition without requiring a numerical bound on its number of components.

Remove any irreducible piece contained in another. The remaining pieces are the maximal irreducible closed subsets, and they are unique up to ordering. To see uniqueness, place one irreducible piece inside a second finite decomposition. Irreducibility forces it into one member there; applying the same reasoning in the opposite direction and using maximality identifies the two pieces. Algebraically these components correspond to minimal primes of the reduced coordinate ring, not to every prime ideal or every intermediate closed subvariety.

Irreducible sets are connected, but connected sets may contain several irreducible components meeting at points. Two intersecting affine lines form the standard example. Connectedness only rules out a decomposition into disjoint nonempty open-and-closed pieces; it does not prohibit intersecting closed pieces. A factorization of a plane equation into distinct irreducible factors gives its reduced curve components, while powers of those factors alter the equation quotient without adding new underlying components. Counting repeated factors as new geometric pieces confuses multiplicity with support.

Primary decomposition in a nonreduced quotient can retain embedded associated primes and thicker component structure. Passing to the radical keeps only the geometric support and its minimal prime components. That distinction matters when a later tangent computation appears larger than expected: it may be responding to a genuine singular intersection, or to nilpotent structure in the chosen equation scheme. Before interpreting a decomposition, decide whether the task concerns the reduced algebraic set or the entire quotient ring, and retain the corresponding ideal consistently.