What is an irreducible variety? Give an example of a reducible one.
Solution
A nonempty topological space \(X\) is irreducible if whenever \[ X=Y\cup Z \] with \(Y,Z\) closed in \(X\), then \(X=Y\) or \(X=Z\). Thus an algebraic variety is irreducible when it cannot be written as the union of two proper closed subvarieties.
For an affine algebraic set \(V(I)\subseteq k^n\), over an algebraically closed field \(k\), one has \[ V(I)\text{ irreducible}\iff I(V(I))=\sqrt I\text{ is prime}. \] Indeed, if \(\sqrt I\) is prime and \[ V(I)=V(J)\cup V(K)=V(JK), \] then \(JK\subseteq\sqrt I\), so primality gives \(J\subseteq\sqrt I\) or \(K\subseteq\sqrt I\), hence one of the two closed subsets is all of \(V(I)\). The converse is obtained by applying irreducibility to \(V(I+(fg))=V(I+(f))\cup V(I+(g))\).
A basic reducible example is \[ V(xy)\subseteq\mathbb A_k^2. \] Since \[ V(xy)=V(x)\cup V(y), \] it is the union of the two coordinate axes, both proper closed subsets. Equivalently, \((xy)\) is not a prime ideal.