# Ideals in $R= \mathbb Z[\sqrt{-5}]/(3)$

I know that there are 9 elements in R, namely $$a+b\sqrt{-5}$$ with $$a,b\in {0,1,2}$$. Now I'm trying to find nontrivial ideals of R, and my book tells me that $$(1+\sqrt{-5})=\{0,1+\sqrt{-5},2+2\sqrt{-5}\}$$. However, I'm confused as to why $$(1+\sqrt{-5})\sqrt{-5}=\sqrt{-5}+-5\equiv 2+\sqrt{-5}$$ is not in this ideal.

• $-5\equiv 1 (mod 3)$ – Mustafa Feb 11 at 9:35
• I love you my friend – davidh Feb 11 at 9:38