my task is following
Let $I=\{a+b\sqrt{10}:13\mid2a-b\}$ be a subset of the ring $\mathbb{Z}[\sqrt{10}]$. Decide if $I$ is an ideal of $\mathbb{Z}[\sqrt{10}]$ and if so, decide if it is principal, prime or maximal.
I've proved that $I$ is ideal indeed (addition is trivial and it is closed under multiplication because $13\mid2a-b\ \Leftrightarrow\ 13\mid a-20b$).
But I have problem with the properties of $I$. I know that $I$ is prime/maximal iff $\mathbb{Z}[\sqrt{10}]/I$ is an integral domain / a field. I don't know, how to continue — $I$ is kernel of some ring homomorphism, but I have no idea how to use it. I suppose we would like to use something like $f:\mathbb{Z}[\sqrt{10}]\to\mathbb{Z}_{13},a+b\sqrt{10}\mapsto[2a-b]_{13}$ but is this even a map?… Edit: oh, of course $f$ is a map, since the element $a+b\sqrt{10}$ represents only itself…
Best Answer
If you have proved that $I$ is an ideal, then you have proved that the map $$f:\mathbb{Z}[\sqrt{10}]\to\mathbb{Z}_{13} \mbox{ given by } a+b\sqrt{10}\mapsto[2a-b]_{13} $$ is a ring homomorphism with kernel $I$.
By considering the elements with $a=0$, it is clear that $f$ is surjective.
What does this tell you about $f$ ?