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The spectrum $\mathrm{Spec}A$ of a commutative ring $A$ is the set of its prime ideals, with closed sets $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$ and a structure sheaf whose sections locally look like fractions.

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.

If $L,L^{\prime }$ are field extensions of $K$, choose a $K$-basis of $L$ containing $1$. After tensoring with $L^{\prime }$, that basis expresses $L{\otimes }_K{L}^{\prime }$ as a nonzero direct sum of copies of $L^{\prime }$, and $1\otimes 1$ is nonzero. A prime ideal of this nonzero ring supplies a point used in the point-lifting property of a scheme fibre product.

Every ideal of a Dedekind domain is generated by at most two elements. Given a nonzero ideal $I$ and any prescribed nonzero $a\in I$, the Chinese remainder theorem supplies $b\in I$ with $v_P((b))=v_P(I)$ at every prime ideal dividing $(a)$. The prime-ideal valuation in a Dedekind domain then gives $I=(a,b)$.

Every nonzero proper ideal of a Dedekind domain has a unique factorization, up to reordering, as a finite product of positive powers of distinct nonzero prime ideals. The unit ideal is the empty product. This is distinct from unique factorization of elements.

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.

In $\mathbb{Z}$, the saturated multiplicative subset $\{-1,1\}$ has complement equal to the union of $(p)$ over all positive primes. No finite subcollection of prime ideals covers this complement: a prime outside the finite list is a nonunit not in any listed ideal.

Given finitely many distinct places of a number field $v_1,\dots ,v_r$, the diagonal image of $K$ is dense in ${\prod }_i{K}_{v_i}$. One can therefore simultaneously prescribe finite congruences and real signs. This is useful in the ray class exact sequence and in finding representatives of an ideal class prime to a prescribed finite set of prime ideals.