Let $R$ be the subring of the real numbers such that $R=\left\{a+b\sqrt{2}:a,b \in \Bbb Z\right\}$ Let $M$ be the ideal in $R$ given by $M=\left\{a+b\sqrt{2}:\text{ a and b are divisible by 5}\right\}$ Prove that $M$is a maximal ideal of $R$.
I was thinking that suppose $M \subseteq P\subseteq R$ and prove that $1\in P$ but I don't know how to prove that. Hope somebody can help me with this