For questions related to primary decomposition. In mathematics, every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals).
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals).
Let $R$ be a Noetherian commutative ring. An ideal $I$ of $R$ is called primary if it is a proper ideal and for each pair of elements $x$ and $y$ in $R$ such that $xy$ is in $I$, either $x$ or some power of $y$ is in $I$; equivalently, every zero-divisor in the quotient $R/I$ is nilpotent. The radical of a primary ideal $Q$ is a prime ideal and $Q$ is said to be $\mathfrak {p}$-primary for $\displaystyle {\mathfrak {p}}={\sqrt {Q}}$.
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