TITLE: Maximal Ideal of a ring.
QUESTION [5 upvotes]: Suppose if $M$ is maximal ideal of $R$ and $a \notin M$, than what can we say about $ M+(a)$?
I studied this from book: The ideal $M + (a)$ contains $M$ so, since $M$ is maximal, this ideal must be all of $R$.
How?

REPLY [8 votes]: $M\subsetneqq M+(a)$, since $a\in\big(M+(a)\big)\setminus M$, so $M+(a)$ is an ideal properly containing $M$. But $M$ is a maximal ideal, so by definition the only ideal properly containing $M$ is the whole ring $R$. Therefore $M+(a)=R$.