TITLE: Sum of two maximal ideal is necesarily maximal ideal?
QUESTION [5 upvotes]: Sum of two prime ideal is not necessarily a prime ideal. Example in the ring $\mathbb{C}[x, y]$ with prime ideals $P = (x^2 + y^2 − 1)$ and $Q = (x)$, but $P+Q$ is not prime ideal. My question is sum of two maximal ideal is necessarily a maximal ideal?

REPLY [7 votes]: If $M, N$ are two distinct maximal ideals then there exists $x \in M$ wlog such that $x \notin N$.  This means that $x \in M + N$ and since $N \subset M + N \subset A$ and $M + N \neq N$ we must have $M + N = A$ the whole ring.