Let $K$ be a field, $S=K[X,Y]$ the polynomial ring in two variables and consider the ideal $M=\langle X,Y\rangle$ (ideal generated by $X$ and $Y$). Show that $M$ is a union of strictly smaller prime ideals.
2026-04-12 18:51:46.1776019906
Decomposition of a maximal ideal as a union of smaller prime ideals
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Hint. The height one prime ideals in $K[X,Y]$ are principal and generated by irreducible polynomials.