Let V (resp. D) be a valuation domain (resp. SFT Prüfer domain), I a proper ideal, and V̂ (resp. D̂) be the I-adic completion of V (resp. D). We show that (1) V̂ is a valuation domain, (2) Krull dimension of V̂ = dimV I+1 if I is not idempotent, V̂ ∼= V I if I is idempotent, (3) dim D̂ = dimD I + 1, (4) D̂ is an SFT Prüfer ring, and (5) D̂ is a catenarian ring. Throughout this paper, all rings are assu...

Let $(R,underline{m})$ be a commutative Noetherian local ring and $M$ be a non-zero finitely generated $R$-module. We show that if $R$ is almost Cohen-Macaulay and $M$ is perfect with finite projective dimension, then $M$ is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient condition on $M$ to be an almost Cohen-Macaulay module, by using $Ext$ functors.

The six-membered thio-pyran ring in the title compound, C(24)H(19)BrS, adopts an approximate envelope conformation, with the S atom displaced by 0.26 (1) Å and the 2-methyl-ene C atom by -0.54 (1) Å from the plane of the other four sp(2)-hydridized C atoms. The methyl substituent on the methyl-ene carbon lies in a pseudo-axial position with the phenyl ring in a pseudo-equatorial position.

The six-membered thio-pyran ring in the title compound, C(26)H(24)S, adopts a boat conformation, with the S atom displaced by 0.478 (2) Å and the 3-methyl-ene C atom by 0.644 (2) Å from the plane of the other four sp(2)-hydridized C atoms. The methyl group on the methyl-ene carbon lies in a pseudo-equatorial position and the phenyl ring in a pseudo-axial position.

Let V be a valuation domain of rank one with quotient field K. We study the set extensions to rational functions K(X) induced by pseudo-convergent sequences K from topological point view, endowing this either Zariski or constructible topology. In particular, we consider two subspaces prescribed breadth pseudo-limit. give some necessary conditions for space metrizable (under topology) in terms v...

We partially generalize the known results on dp-minimal fields to dp-finite fields. prove a dichotomy: if K is sufficiently saturated expansion of field, then either has finite Morley rank or non-trivial Aut(K/A)-invariant valuation ring for some small set A. In positive characteristic case, we can even obtain henselian ring. Using this, classify pure

Given a valuation on the function field k(x, y), we examine the set of images of nonzero elements of the underlying polynomial ring k[x, y] under this valuation. For an arbitrary field k, a Noetherian power series is a map z : Q → k that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on k(x, y). Although the value groups correspondi...