The SFT property and the ring R((X))

An ideal I is called an SFT-ideal if there exist a natural number n and a finitely generated ideal Jsubset of or equal toI such that xnset membership, variantJ for each xset membership, variantI. An SFT-ring is a ring such that every ideal is an SFT-ideal. For a commutative ring D, let D((X)) be the power series ring D[[X]] localized at the power series with unit content ideal. We show that for a Prüfer domain D, all the prime ideals of D((X)) are formally extended from D if and only if D((X)) is SFT if and only if D is SFT.