Invertibility of elements in $A[x]$ with coefficients in the Jacobson radical

It does not have to be the case that $\mathfrak{R}((x) + \cdots + (x^n))$ is finite, so Nakayama is not applicable here in general.

There is a second, much more serious error, however: for the argument to work, one would need that $M = A + \mathfrak{R} \cdot \mathfrak{R}((x) + \cdots + (x^n))$, but that does not follow from $M = A + \mathfrak{R}((x) + \cdots + (x^n))$!