Let R be a finite commutative ring with 1.Let a,b∈ R such that (a)+(b)=R and M be any maximal ideal of R with b∉M. Then is some x∈ R such that a+xb∉M

Question

Let $R$ be a finite commutative ring with $1$.Let $a , b\in$ $R$ such that ($a$)+($b$)=$R$ and $M$ be any maximal ideal of $R$ with $b\not\in M$.Then I have to prove that there is some $x\in$ $R$ such that $a+xb\not\in$$M$

This is a part of a problem,may be not all the conditions will be needed.I have been trying it for last few days but I am not getting it anyway.Please help.Thank you.

Answer 1

We don't need $(a)+(b)=R$ and we don't need that $R$ is finite.

Since the question is really only about membership in $M$ (or not in $M$), we may as well consider $R/M$, which is a field. The question becomes:

If $F$ is a field with elements $a,b$ such that $b \neq 0$, then there is some $x \in F$ such that $a+xb \neq 0$.

But this is trivial, just $x=\frac{1-a}{b}$, then $a+xb=1$.