Prove that $m^2$ is primary

Question

Let $m$ be a maximal ideal. I'm having a hard time proving that $m^2$ is primary.

Let ${xy\in m^2}$ so $xy=t_{1}s_{1}+...+t_{n}s_{n}$ where the $t_{i},s_{i}$ are in $m$.

Answer 1

in the book "Steps in Commutative Algebra" (Second edition) by "R. Y. Sharp" you see

this is a direct proof that summarize the main proof above:
If $ab \in m^2$ and $b \notin m$ then $Rb+m=R$ so $1=rb+ t$ ; $t \in m$ , $r \in R$ . hence $a=a1=a(rb+ t)= rab+ at \in m^2 $ (note that $ ab \in m^2 $ and $a,t \in m$)