Methods to show an ideal in the ring of integers $\mathcal{O}_K$ is a proper ideal I was wondering if there were general "tactics" to show if an ideal in the ring of integers $\mathcal{O}_K$, where $K/\mathbb{Q}$ is a number field of degree $n$.
For example, consider $\mathcal{O}_K = \mathbb{Z}[\sqrt{-5}]$ and $P = (2,1+\sqrt{5})$. One method to show this is a proper idea is to argue that if $a + b\sqrt{-5} \in P,$ then $a+b\sqrt{-5} = 2x + y(1+\sqrt{-5})$ for integers $x,y$ and comparing coefficients, we get $a - b \equiv 0 \bmod 2,$ so $P$ cannot be the whole ring. However, this method seems quite laborious, and this may just be a special case when modding out by a prime actually works.
For me, the following isomorphism seems quite intuitive: as $\mathbb{Z}[\sqrt{-5}] \cong \mathbb{Z}[x]/(x^2+5),$ we should have that 
\begin{align*}
\mathbb{Z}[\sqrt{-5}]/(2,1+\sqrt{5}) &\cong \mathbb{Z}[x]/(x^2+5,2,1+x) \\
&\cong \mathbb{Z}/(6,2) \text{ by mapping } x \mapsto -1 \\
&\cong \mathbb{Z}/2\mathbb{Z}
\end{align*}
so the ideal $P$ is indeed proper. But I can't seem to set up an explicit isomorphism at the moment, so I was wondering if this works generally?
Furthermore, are there any useful techniques one might use to show an ideal is indeed proper?
 A: A "proper" ideal is any ideal "which is strictly smaller than the whole ring," right?
Then it's enough to show that the given ideal does not contain 1. If the ideal contains a unit, it must also contain 1 and must therefore be the whole ring.
In your example, we have $\mathfrak P = \langle 2, 1 + \sqrt{-5} \rangle$, which is simply the set of all numbers in this domain of the form $2x + y(1 + \sqrt{-5}) = a + b \sqrt{-5}$. The contribution of $2x$ to $a$ and $b$ is even. We can certainly choose $y$ so that $a = 1$, but then $b \neq 0$ like we want, since in fact $b$ must be odd for $a$ to be odd.
I can't set up an isomorphism either, that's a deficiency of mine, but it's kind of overkill if you just want to show that an ideal is proper.
A: To know is an ideal $\mathfrak{a}$ is proper or not isn't easier if we just compute the size of $\mathcal{O}_K/ \mathfrak{a}$ thinking of $\mathcal{O}_K$ as a lattice?.
As long as  you know a $\mathbb{Z}$-basis for $\mathcal{O}_K$ say $\{ 
\gamma_1, \ldots,  \gamma_n \}$ and you're given $\mathfrak{a}$ is terms of $\mathcal{O}_K$-generators say $  (\beta_1,\ldots,\beta_m)$ you can esaily find a $\mathbb{Z}$-basis $\{ 
\alpha_1, \ldots,  \alpha_n \}$ for $\mathfrak{a}$  ( reducing the $\mathbb{Z}$-generators of $\mathfrak{a}$ $ \{\beta_i \gamma_j \}$ to a $\mathbb{Z}$-basis, for example by hermite form). Then you simply write $\alpha_i=\sum a_{ij} \cdot \gamma_j$  and  $ | \mathcal{O}_K/ \mathfrak{a} |=|\text{det}(a_{ij})|$ would be $1$ iff $\mathcal{O}_K=\mathfrak{a}$.
In your example $\{1, \sqrt{-5} \}$ is a $\mathbb{Z}$-basis for $\mathcal{O}_K$,  $\{2,2\sqrt{-5},1+\sqrt{-5}, (1+\sqrt{-5})\sqrt{-5} \}$ reduces to the $\mathbb{Z}$-basis $  \{1+\sqrt{-5},2 \sqrt{-5}\}$ for $P$, because the hermite form of
$$
\begin{bmatrix}
    2       & 0  \\
    0       & 2 \\
    1       & 1\\
   -5       & 1 
\end{bmatrix}
$$
is the matrix
$$
\begin{bmatrix}
    1      & 1  \\
    0       & 2 \\
    0       & 0\\
   0      & 0 
\end{bmatrix}
$$
and so $ | \mathcal{O}_K/ P |=\text{det}\begin{bmatrix}
    1      & 1  \\
    0       & 2  
\end{bmatrix}$$=2\neq 1$.
A: I'm sure there is a much better way, but yes it works generally, assuming you know the multiplication law in $\mathcal{O}_K$ seen as a free $\mathbb{Z}$ module.
Find $r \in I \cap \mathbb{Z}$, list all the elements of the finite ring $$R_0 = \mathcal{O}_K/(r)$$
then write $I = (u_1,\ldots,u_n)$ and list all the elements of the finite quotient rings $$R_1= R_0/(u_1), \quad R_2= R_1/(u_2), \quad R_{m+1} = R_m/(u_{m+1})$$ You'll get $$I = \mathcal{O}_K \qquad \Longleftrightarrow \qquad \mathcal{O}_K / I = R_n = \{0\}$$

The more general setup is $$\mathcal{O}_K = \mathbb{Z}[X_1,\ldots,X_k]/J$$ for some ideal $J$, and we want to know if $$\mathcal{O}_K= I \qquad \Longleftrightarrow \qquad (I,J) = \mathbb{Z}[X_1,\ldots,X_k]$$
there is an algorithm for that using Gröbner basis 
