$R$ commutative ring with unity. Prove if $R/M$ is a field $\implies M$ is a max ideal.
Let us propose the opposite: that M isn't the max ideal. So $$I \triangleleft R, M\nsubseteq I$$ (How this is opposite of M not being the max ideal i don't understand). Now we take $a\in R$ and $b \in I|M$ $; R|M$ field $\implies \exists x+M; a+M=(x+M)(b+M)$ $$a+M=bx+M$$ $$a+M-(bx-M)=M \implies a-bx \in M;$$ $$ **a - bx \in I;b\in I \implies a\in I \implies I=R**$$
I don't understand the highlighted and the brackets before.. Can anyone help ?