Primääridekompositioon

Primääridekompositioon, also known as primary decomposition, is a fundamental concept in commutative algebra and algebraic geometry. It deals with the representation of ideals in a ring as an intersection of simpler ideals. Specifically, in a Noetherian ring, any ideal can be expressed as the intersection of a finite number of primary ideals. This decomposition is crucial for understanding the structure of ideals and the geometry of algebraic varieties.

A primary ideal $Q$ in a commutative ring $R$ is defined by the property that if $ab \in Q$ and $a \notin Q$, then $b^n \in Q$ for some positive integer $n$. The radical of a primary ideal $Q$, denoted by $\sqrt{Q}$, is a prime ideal. The decomposition of an ideal $I$ into primary ideals is analogous to the prime factorization of integers. For instance, just as an integer can be uniquely factored into prime powers, an ideal can be decomposed into primary ideals.

The primary decomposition of an ideal is not unique in general. However, there are unique aspects associated with it. The set of prime ideals that are radicals of the primary ideals in a decomposition is unique. These are called the associated primes of the ideal. Furthermore, the primary ideals that correspond to minimal primes are unique. This uniqueness allows for the study of the geometry of the zero set of the ideal. The minimal primes correspond to the irreducible components of the algebraic variety defined by the ideal, and the non-minimal primes correspond to embedded components. Primary decomposition provides a powerful tool for analyzing these geometric structures.