For a commutative Noetherian ring R, if R[[x]] is a unique factorization domain, then R[[x,y]] is also a unique factorization domain.
On Landweber`s unique factorization problem
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abstract
We solve a long-standing open problem, posed by Landweber in 1974: Let $R = K[x_1, x_2, . . . ]$ be the ring of polynomials in countably many variables over a field $K$. Is the formal power series ring $R[[t]]$ a unique factorization domain? We prove that it is. The proof is based on a new general result in commutative algebra: If $R$ is a Krull domain, and $f \in R[[t]]$ is irreducible, then $f$ is irreducible modulo a finite power of $t$.
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The Noetherian Case of Bayart's Power-Series Question
For a commutative Noetherian ring R, if R[[x]] is a unique factorization domain, then R[[x,y]] is also a unique factorization domain.