Maximal ideal in ring of power series

Question

If $R$ is a commutative ring with identity we know that the maximal ideals of the ring of power series over $R$ have the form $M’=(M,x)$ where $M$ is a maximal ideal of $R$. Do you have a counterexample that shows that if $R$ doesn’t have an identity then the theorem doesn’t hold? I really don’t have any idea where or how to start. Reference: Burton’s “First course in ring and ideals” page 117 theorem 7-4

Answer 1

I don't know for sure if this suits your needs or not, but if $R=2\mathbb Z/4\mathbb Z$ and $M=(2x)\lhd R[[x]]$, then $R[[x]]/M\cong R$ has two elements, so $M$ is maximal (in the sense you specified in the comments.)