Question:
Let $R = \{a + bi | a,b \in \Bbb Z \}$, let $M = \{x(2 + i) | x \in R\}$. Prove M is a maximal ideal of R.
I just started learning about ideals so I apologize for asking a basic question, please describe steps as simply as possible.
After reading online and related questions on here, I was trying to first prove M is an ideal and then prove R/M is a field implying M is a maximal ideal. Is this misguided? And, if not, how would I go about doing step two?
It's not misguided. You could prove that $R/M$ is a field by thinking about what field it would be. Hint: rewrite $R = \mathbb{Z}[i]$ as $\mathbb{Z}[T]/(T^2+1)$, which makes the quotient easier to calculate.
You could also take the easier approach of calculating the number of elements of $R/M$, which tells you quite a bit about its structure.