SFT stability via power series extension over Prüfer domains

Abstract

A ring D is called an SFT ring if for each ideal I of D, there exist a natural number k and a finitely generated ideal ${J\subseteq{I}}$ such that a ^k∈J for each a∈I. We show that the power series ring ${D[\![x_1,\ldots, x_n]\!]}$ over an SFT Prüfer domain D is again an SFT ring even if D is infinite-dimensional. From this, it follows that every ideal-adic completion of D is also an SFT ring. We also show that ${D[\![x_1,\ldots, x_n]\!]_{D{\setminus}(0)}}$ is an n-dimensional regular ring.