Exercise 12, § 2

Let $A$ be a ring, $M$ an $A$-module and $N$ a submodule of $M$. (a) For $\operatorname{Ass}_f(M/N)$ (§ 1, Exercise 17) to be reduced to a single element $p$, it is necessary and sufficient that $N \neq M$ and that, for all $a \in A$, the homothety with ratio $a$ on $M/N$ be injective or almost nilpotent; then $p = r_M(N)$ (Exercise 11) and we also say that $N$ is primay (or $p$-*primay*) in $M$. For every submodule $M'$ of $M$ such that $N \subset M'$ and $N \neq M'$, $N$ is then $p$-primary in $M'$. (b) If $r_M(N)$ is a maximal ideal of $A$, show that $N$ is primary in $M$. In particular, every ideal $q$ of $A$ which is only contained in a single prime ideal $m$ (necessarily maximal) is $m$-primary. (c) Let $A$ be an integral domain and $x$ an element of $A$ such that $p = Ax$ is prime; then, for every integer $n \geq 1$, $Ax^n$ is $p$-primary. Show that this result does not extend to the case where A is not an integral domain (take $A = B/b$, where $B = K[X, Y]$ (K being a field), $b = BX^2 + BXY$).

(d) If A is an absolutely flat ring (Chapter I, § 2, Exercise 17), the primary ideals of A are identical with the prime ideals of A.

(e) Give an example of a Z-module M such that {0} is p-primary (for a prime number $p$) but that for no $a \neq 0$ in $\mathbf{Z}$ is the homothety $a_M$ nilpotent (cf. Chapter II, § 2, Exercise 3).