I work on exercise 23, page 295 from "commutative algebra, bourbaki". One direction of this problem is easy. To show that the conditions are sufficient, there is hints for solving. I could understand all of this hints except "maximality of $G_n$ among the submodules $G$ containing $G_{n-1}$ and such that $F=H_n
\cap G$". I can't verify this. Can anyone help me? thanks a lot.
The exercise is this:
Let $A$ be a ring. An $A-$module $E$ is laskerian if it is finitely generated and every submodule of $E$ has a primary decomposition in $E$.
for a finitely generated $A-$module $E$ to be laskerian, it is necessary and sufficient that it satisfy the two following axioms:
(1) for every submodule $F$ of $E$ and every prime ideal $p$ of $A$, the saturation of $F$ with respect to $p$ in $E$ is of the form $F:(a)$ for some $a\notin p$.
(2) for every submodule $F$ of $E$, every decreasing sequence $(sat_{S_n}(F))$ is satationary.
Hint: to show that the conditions are sufficient, prove first using (1) and exercises 18(a) and 22(a), that, for every submodule $F$ of $E$, there exist a submodule $Q$ of $E$ which is primary for some ideal $p\supset F:E$ and a submodule $G=F+aE$, where $a\notin p$, such that $F=Q \cap G$. then argue by reductio ad absurdum: show that there would exist an infinite sequence $(Q_n)$ of $p_n$-primary submodules of $E$ and a strictly increasing $(G_n)$ of submodules of $E$, such that, if we write $H_n=Q_1 \cap ... \cap Q_n$: (1) $F=H_n \cap G_n$, (2) $G_n$ is maximal among the submodules $G$ containing $G_{n-1}$ and such that $F=H_n \cap G$. (3) ...