Anti-archimedean rings and power series rings

S-Noetherian Properties of Composite Ring Extensions

“…⇐ We first show that D + X 1 X n E X 1 X n is S-Noetherian, where X 1 X n are indeterminates over E. Since D is S-Noetherian and E is an S-finite D-module, E is an S-Noetherian ring by Lemma 3.5 (2). Also, since S is an anti-Archimedean subset of E consisting of nonzerodivisors, E X 1 X n is S-Noetherian [1, Proposition 9].…”

Chain conditions on composite Hurwitz series rings

“…Let R be a commutative ring with identity and S a (not necessarily saturated) multiplicative subset of R. We say that S is anti-Archimedean if T n 1 s n R \ S ¤ ; for every s 2 S. We also say that an integral domain R is an anti-Archimedean domain if T n 1 a n R ¤ 0 for each 0 ¤ a 2 R (see [16]). Thus R is an anti-Archimedean domain if and only if R n f0g is an anti-Archimedean subset of R. Clearly, every multiplicative subset consisting of units is anti-Archimedean.…”

Integral domains with almost integral proper overrings

“…All rings considered below are (commutative integral) domains. Let R be a domain, with complete integral closure R Ã and quotient field K. A key motivation for this work is the observation that if R is a conducive domain (in the sense of [7]) which is not a field, then: R has no height 1 prime ideal D R is not a G-domain D R Ã K D R is pointwise non-Archimedean (in the sense of [6], also termed anti-Archimedean in [1]). In this introduction, we sketch how the preceding observation relates to the overall plan of this paper.…”

“…(The ªpointwise non-Archimedeanº concept was re-discovered in [1], where it was termed ªanti-Archimedean.º It is curious that although [6] and [1] introduced this concept for different purposes, both of those purposes involved formal power series.) A key fact [1, Proposition 2.4] is that R is pointwise non-Archimedean if and only if P r o o f. Assume that each proper nontrivial overring of R is almost integral over R; equivalently, that T 7 R Ã for each proper nontrivial overring T of R. Now, either R Ã K or R Ã j K. In the former case, R is pointwise non-Archimedean, by [1,Proposition 2.4]. In the latter case, the hypothesis ensures that R Ã contains each overring of R other than K, whence R is maximized, with maximum overring R Ã .…”

“…The first of these equivalences follows directly from the definition of R Ã and is essentially the proof of[1, Proposition 2.4]; the second equivalence is reminiscent of ªsimply conduciveº studies, such as the proof of [7, Lemma 4.2 (ii)].Recall from[7] that R is called a maximized domain if R has an overring T j K such that T contains each overring of R other than K. If it exists, such a ring T is termed a (the) maximum overring of R. Maximized domains form a subclass of the class of G-domains and are characterized in [7, Proposition 4.1]. Theorem 2.3.…”