The preparation theorem and the freeness of $A[[X]]/I$

Abstract:

Let $I$ be a nonzero ideal of $A[[X]]$, the ring of formal power series over a commutative Noetherian ring $A$. These are equivalent: (i) $I$ is generated by a single series $f = {a_0} + {a_1}X + \ldots$ such that for some $s,\;{a_s}$ is a unit, the first $s$ coefficients ${a_0}, \ldots ,{a_{s - 1}}$ of $f$ lie in the Jacobson radical of $A$ and $A$ is complete in the adic topology defined by the ideal they generate. (ii) $A[[X]]/I$ is a free $A$-module.