10 - Structure Theorem in Primary Decomposition Theorem Form

Structure Theorem in Primary Decomposition Theorem Form

Let $R$ be a Euclidean Domain
Let $M$ be a finitely generated $R$-module

Then there are irreducibles $p_1,\cdots p_k\in R$ and integers $s,r_i$ where $1\le i\le k$ such that

$$M\cong \underset{i=1}{\overset{k}{⨁}}(R/p_i^{r_i}R)\oplus R^s$$

where pairs $(p_i,r_i)$ are uniquely determined up to units