We study rings over which an analogue of the Weierstrass preparation theorem holds for power series. We show that a commutative ring R admits a factorization of every power series in R[[x]] as the product of a polynomial and a unit if and only if R is isomorphic to a finite product of complete local principal ideal rings. We also characterize Noetherian rings R for which this factorization holds under the weaker condition that the coefficients of the series generate the unit ideal: this occurs precisely when R is isomorphic to a finite product of complete local Noetherian integral domains.
Beyond this, we investigate the failure of Weierstrass-type preparation in finitely generated rings and prove a general transcendence result for zeros of p-adic power series, producing a large class of power series over number rings that cannot be written as a polynomial times a unit. Finally, we show that for a finitely generated infinite commutative ring R, the decision problem of determining whether an integer power series (with computable coefficients) factors as a polynomial times a unit in R[[x]] is undecidable.
You can find the preprint on arXiv: https://arxiv.org/abs/2504.10725
