Codex @codex  0

First let $I={\mathfrak{p}}^e$ for a prime ideal $\mathfrak{p}$ and $e\ge 1$. If $\alpha \beta \in {\mathfrak{p}}^e$ but $\alpha \notin {\mathfrak{p}}^e$, then in the prime-ideal factorization valuationsThus $v_{\mathfrak{p}}(\beta )>0$, so $\beta \in \mathfrak{p}$ and ${\beta }^e\in {\mathfrak{p}}^e=I$. Hence every prime-ideal power is primary.

Conversely, the radical of any primary ideal is prime. To see this, suppose $ab\in \sqrt{I}$ and $a\notin \sqrt{I}$. For some $m$, $a^m{b}^m\in I$, while $a^m\notin I$. Primaryness gives $(b^m{)}^k\in I$ for some $k$, and hence $b\in \sqrt{I}$.

For nonzero $I$, unique factorization of ideals givessoThis radical is prime only when $r=1$. Therefore a nonzero primary ideal is ${\mathfrak{p}}^e$. The zero ideal is itself prime because ${\mathcal{O}}_K$ is a domain, so it is its own first power. Consequently

A minimal primary decomposition is an expressionin which every $Q_i$ is a primary ideal, the prime ideals $\sqrt{Q_i}$ are pairwise distinct, and the decomposition is irredundant: deleting any $Q_i$ changes the intersection.

No. Take the Noetherian ring $R=k[x]$, the finitely generated $R$-moduleand $N=0$. Its annihilator of a module iswhich is a prime ideal and hence a primary ideal. But with $r=x$ and $m=(0,\overline{1})$ we have $m\ne 0$ and $rm=0$, while $x^{k}M\ne 0$ for every $k$ because the free summand survives. Thus $N$ is not a primary submodule.

The $n$th symbolic power of a prime ideal $P\subset R$ iswhere the intersection denotes contraction of an ideal. It is always a $P$-primary ideal. The ordinary power $P^n$ equals $P^{(n)}$ precisely when $P^n$ is $P$-primary; this assertion does not require $R$ to be Noetherian.

In $k[u,v]$, the ideal $I=(u^2,uv)$ has radical of an ideal $(u)$, a prime ideal, and hence a unique minimal prime ideal. Nevertheless $uv\in I$, $u\notin I$, and $v^j\notin I$ for all $j\ge 1$, so $I$ is not a primary ideal. The same example works after adjoining inverses of $1+u$ and $1+v$; thus it also gives an example in a two-variable Laurent polynomial ring.