Primääridekompositioiksi

Primääridekompositioiksi is a Finnish term that translates to "primary decompositions" in English. This concept arises in abstract algebra, particularly in ring theory and module theory. It is a generalization of the fundamental theorem of arithmetic, which states that every integer greater than one can be uniquely represented as a product of prime numbers.

In the context of rings, a primary decomposition of an ideal $I$ in a commutative ring $R$ expresses $I$ as an intersection of primary ideals. A primary ideal $Q$ is an ideal such that if $ab \in Q$ and $a \notin Q$, then $b^n \in Q$ for some positive integer $n$. The collection of primary ideals in the decomposition is called a primary decomposition.

Similarly, in module theory, a primary decomposition of a submodule $N$ of a module $M$ over a commutative ring $R$ expresses $N$ as an intersection of primary submodules. A primary submodule $P$ of $M$ is a submodule such that if $xm \in P$ for $x \in R$ and $m \in M$, and $x$ is not a zero divisor modulo $P$ (i.e., $xm \in P$ implies $m \in P$ for all $m \in M$), then $m \in P$.

A key aspect of primary decomposition is that while the decomposition itself may not be unique, the associated prime ideals (known as the radical of the ideal or submodule) are uniquely determined. These primes correspond to the "minimal" or "maximal" prime ideals associated with the decomposition. The study of primary decompositions provides valuable insights into the structure of rings and modules, particularly Noetherian rings and modules, where such decompositions are guaranteed to exist.