IJPAM: Volume 23, No. 3 (2005)

$p$-BANACH COMMUTATIVE ALGEBRAS WHOSE
RADICAL IDEALS ARE FINITELY
GENERATED OR CLOSED

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Let $A$ be a commutative unitary and complete $p$-normed algebra, $0 < p \le 1$. Consider the following conditions:

(i) every prime ideal of $A$ is closed;

(ii) every radical ideal of $A$ is closed;

(iii) every radical ideal of $A$ is finitely generated.

(iv) every chain of prime ideals of $A$ is stationary;

(v) every chain of radical ideals of $A$ is stationary;

(vi) every radical ideal of $A$ is the radical of a finitely generated ideal.

Then: (i) $\Longleftrightarrow$ (ii); (ii) $\Longrightarrow $ (v) $\Longrightarrow $ (iv); (ii) $\Longrightarrow $ (v) $\Longrightarrow $ (vi) and (iii) $\Longrightarrow $ (ii).

Received: June 21, 2005

AMS Subject Classification: 32K05, 32K99

Key Words and Phrases: $p$-Banach spaces, topological algebra, topological noetherian algebra, radical ideal

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 23
Issue: 3