Growth and uniqueness of rank
type: Item
instance of: scholarly article
title: Growth and uniqueness of rank
zbMATH Open document ID: 0673.16014
DOI: 10.1007/BF02787226
author: Eli Aljadeff
author: Shmuel Rosset
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let R be a ring. Then R is said to have UR (the unique rank property) if every free R-module has a well-defined rank. The main aim of the paper is to show that R has UR if R has a subring S of the type defined below and R is related to S in a way which generalizes being finitely-generated as an S-module. The ring S is monotone in the sense that if the direct sum of k copies of S embeds as an S-module in the direct sum of n copies of S, then k is less than or equal to n. In particular S could be commutative or a Goldie ring. The connection between R and S is given by a ``growth structure'' which we will not define precisely here, but it involves an increasing filtration of R by free S-modules of finite rank. It is shown that if the corresponding growth is subexponential then R is monotone and hence has UR.
reviewed by: A. W. Chatters
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 4101398
zbMATH Keywords: algebras of sub-exponential growth
zbMATH Keywords: growth exponent
zbMATH Keywords: unique rank property
zbMATH Keywords: direct sum
zbMATH Keywords: Goldie ring
zbMATH Keywords: growth structure
zbMATH Keywords: increasing filtration
zbMATH Keywords: free S-modules
MaRDI profile type: Publication
cites work: Global dimensions of crossed products
cites work: Q3890834
cites work: Q3338434
cites work: A note on curvature and fundamental group
description: scientific article; zbMATH DE number 4101398