I will assume that everything in sight is Noetherian and finitely generated. Then the primary decomposition amounts to a description of the geometric support of the module $M$ if you view it as a sheaf on $\text{Spec}(A)$. The scheme $\text{Spec}(A/\text{Ann}(M))$ is a subscheme of $\text{Spec}(A)$ that, by definition, supports $M$. If $M_i$ N_i$is minimal, then$\text{Spec}(A/P_i)$is an irreducible component of$\text{Spec}(A/\text{Ann}(M))$. In a reduced decomposition, the The module$M/N_i$is also a quotient of$M \otimes (A/P_i^n)$for$n$large enough (and in a reduced decomposition I think)think they are equal), so it is the fiber part of$M$on that irreducible component of its support. If$N_i$is not minimal, then$\text{Spec}(A/P_i)$is an irreducible scheme inside of a component of$\text{Spec}(A/\text{Ann}(M))$. 1 I will assume that everything in sight is Noetherian and finitely generated. Then the primary decomposition amounts to a description of the geometric support of the module$M$if you view it as a sheaf on$\text{Spec}(A)$. The scheme$\text{Spec}(A/\text{Ann}(M))$is a subscheme of$\text{Spec}(A)$that, by definition, supports$M$. If$M_i$is minimal, then$\text{Spec}(A/P_i)$is an irreducible component of$\text{Spec}(A/\text{Ann}(M))$. In a reduced decomposition, the module$M/N_i$is also$M \otimes (A/P_i^n)$for$n$large enough (I think), so it is the fiber of$M$on that irreducible component of its support. If$N_i$is not minimal, then$\text{Spec}(A/P_i)$is an irreducible scheme inside of a component of$\text{Spec}(A/\text{Ann}(M))\$.