Slender domains and compact domains (Q2363344)

Slenderness is a ``discreteness condition'' on a commutative domain \(R\): \(R\) is slender if for every linear \(\phi:\, R^{\mathbb N}\to R\), only finitely many \(\phi(1_n)\neq 0\). The notions of slenderness and (algebraic) compactness are opposites, and in order to study generalizations of these concepts, the authors find it convenient to compare their negations, namely \(R\) is non-slender (n-sl) if there exists \(\phi\) such that \(\phi(1_n)\neq 0\) for all \(n\); nearly compact (nl-c) if there exists \(\phi\) such that for some \(0\neq u\in R,\;\phi(1_n)=u\) for all \(n\); and \(\aleph_0\)-compact \((\aleph_0\)-c) if there exists \(\phi\) such that \(\phi(1_n)=1\) for all \(n\). The implications \((\aleph_0\)-c)\(\Rightarrow \)(nl-c)\(\Rightarrow\)(n-sl) are obvious, and it is the purpose of this paper to study the converses. The main results are: {\parindent=0.7cm\begin{itemize} \item[--] \((\aleph_0\)-c) is equivalent to (nl-s) for Noetherian domains, and this condition holds if and only if \(R\) is local and complete; \item[--] all three conditions are equivalent if in addition \(R\) is one-dimensional; \item[--] (nl-c)\(\Rightarrow\)(n-sl) is irreversible for Noetherian domains of dimension \(\geq 2\), and \((\aleph_0\)-c)\(\Rightarrow \)(nl-c) is irreversible for non-Noetherian domains; \item[--] Domains of countable vector space dimension over a field are slender. \end{itemize}}