Two properties of the power series ring
type: Item
instance of: scholarly article
title: Two properties of the power series ring
zbMATH Open document ID: 0701.13010
DOI: 10.1155/S0161171288000031
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46473
review text: A commutative ring with identity A is called a PF [resp. PP] ring if every principal ideal is flat [resp. principal]. The author proves the following: (1) The power series ring A \(X\) is a PF ring if and only if, for any two countable subsets S and T of A such that \(S\subset ann_ A(T)\), there exists \(c\in ann_ A(T)\) such that \(bc=b\) for all \(b\in S\). (2) \(AX\) is a PP ring if and only if A is a PP ring in which every increasing chain of idempotents in A has a supremum which is an idempotent. In (2) the supremum is relative to the partial order given by \(a\leq b\leftrightarrow a^ 2=ab\).
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13C11
Mathematics Subject Classification ID: 13C10
zbMATH DE Number: 4148252
zbMATH Keywords: flat ideal
zbMATH Keywords: principal ideal
zbMATH Keywords: power series ring
zbMATH Keywords: PF ring
zbMATH Keywords: PP ring
author: H. Al-Ezeh
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4148252