Journal of Pure and Applied Algebra 204 (2006) 270 – 279
The SFT property and the ring R((X))
, Byung Gyun Kang
Department of Mathematics, University of Science, Tunis, Tunisia
Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea
Received 22 April 2004; received in revised form 24 March 2005
Available online 1 June 2005
Communicated by C.A. Weibel
An ideal I is called an SFT-ideal if there exist a natural number n and a ﬁnitely generated ideal
J ⊆ I such that x
∈ J for each x ∈ I . 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.
© 2005 Elsevier B.V. All rights reserved.
MSC: 13A15; 13F05; 13F25
In this paper, all the rings are commutative rings with identity. The dimension of a ring
means the Krull dimension. All the terms and notation are standard as in .
An Ideal I is called an SFT-ideal if there exist a natural number n and a ﬁnitely generated
ideal J ⊆ I such that x
∈ J for each x ∈ I. An SFT-ring is a ring whose every ideal
is an SFT-ideal. For properties about SFT rings the readers are referred to [4,5]. Let D be
a ring. We denote by N the set of power series over D with unit content. Let D((X)) be
D[[X]] localized at N. For an ideal Q of D[[X]] (resp., D((X))), we say that Q is formally
extended from D if Q = P[[X]] (resp., P[[X]]
) for some ideal P in D. Our goal is to
prove the following result.
E-mail addresses: firstname.lastname@example.org (S. Elaoud), email@example.com (B.G. Kang).
Supported by Korea Research Foundation Grant (KRF-2003-041-C00008).
0022-4049/$ - see front matter © 2005 Elsevier B.V. All rights reserved.