On chains of prime submodules

In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structur...

Full description

Saved in:
Bibliographic Details
Published in:Israel journal of mathematics 2002-01, Vol.127 (1), p.131-155
Main Authors: Man, Shing Hing, Smith, Patrick F.
Format: Article
Language:English
Subjects:
Chains
Domains
Mathematics
Modules
Rings (mathematics)
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by cdi_FETCH-LOGICAL-c257t-b743f414cba616be975cff012054fe7f673398a7b242e0f4a3840119200d4bdd3
cites cdi_FETCH-LOGICAL-c257t-b743f414cba616be975cff012054fe7f673398a7b242e0f4a3840119200d4bdd3
container_end_page 155
container_issue 1
container_start_page 131
container_title Israel journal of mathematics
container_volume 127
creator Man, Shing Hing
Smith, Patrick F.
description In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structure of the underlying ring. The dimension of finitely generated modules over a Dedekind domain is also studied. By examining the structure of prime submodules, a formula for the dimension of a free module of finite rank, over a Noetherian one-dimensional domain, is obtained.
doi_str_mv 10.1007/BF02784529
format article
fullrecord <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2786555462</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2786555462</sourcerecordid><originalsourceid>FETCH-LOGICAL-c257t-b743f414cba616be975cff012054fe7f673398a7b242e0f4a3840119200d4bdd3</originalsourceid><addsrcrecordid>eNpFj0FLAzEQhYMouFYv3oUFb8LqzCST7B61tCoUetFzSHYTbGl3a9I9-O9tqeDpXT7ee58QtwiPCGCeXuZAplZMzZkokDVXNSOeiwKAsCI0dCmucl4DsDQoC3G37Mv2y636XA6x3KXVNpR59NuhGzchX4uL6DY53PzlRHzOZx_Tt2qxfH2fPi-qltjsK2-UjApV651G7UNjuI0RkIBVDCZqI2VTO-NJUYConKwVIDYE0CnfdXIi7k-9uzR8jyHv7XoYU3-YtAcdzcxK04F6OFFtGnJOIdrjX5d-LII96tt_ffkLF8RJcQ</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2786555462</pqid></control><display><type>article</type><title>On chains of prime submodules</title><source>Springer Nature</source><creator>Man, Shing Hing ; Smith, Patrick F.</creator><creatorcontrib>Man, Shing Hing ; Smith, Patrick F.</creatorcontrib><description>In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structure of the underlying ring. The dimension of finitely generated modules over a Dedekind domain is also studied. By examining the structure of prime submodules, a formula for the dimension of a free module of finite rank, over a Noetherian one-dimensional domain, is obtained.</description><identifier>ISSN: 0021-2172</identifier><identifier>EISSN: 1565-8511</identifier><identifier>DOI: 10.1007/BF02784529</identifier><language>eng</language><publisher>Heidelberg: Springer Nature B.V</publisher><subject>Chains ; Domains ; Mathematics ; Modules ; Rings (mathematics)</subject><ispartof>Israel journal of mathematics, 2002-01, Vol.127 (1), p.131-155</ispartof><rights>The Hebrew University Magnes Press 2002.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c257t-b743f414cba616be975cff012054fe7f673398a7b242e0f4a3840119200d4bdd3</citedby><cites>FETCH-LOGICAL-c257t-b743f414cba616be975cff012054fe7f673398a7b242e0f4a3840119200d4bdd3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Man, Shing Hing</creatorcontrib><creatorcontrib>Smith, Patrick F.</creatorcontrib><title>On chains of prime submodules</title><title>Israel journal of mathematics</title><description>In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structure of the underlying ring. The dimension of finitely generated modules over a Dedekind domain is also studied. By examining the structure of prime submodules, a formula for the dimension of a free module of finite rank, over a Noetherian one-dimensional domain, is obtained.</description><subject>Chains</subject><subject>Domains</subject><subject>Mathematics</subject><subject>Modules</subject><subject>Rings (mathematics)</subject><issn>0021-2172</issn><issn>1565-8511</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2002</creationdate><recordtype>article</recordtype><recordid>eNpFj0FLAzEQhYMouFYv3oUFb8LqzCST7B61tCoUetFzSHYTbGl3a9I9-O9tqeDpXT7ee58QtwiPCGCeXuZAplZMzZkokDVXNSOeiwKAsCI0dCmucl4DsDQoC3G37Mv2y636XA6x3KXVNpR59NuhGzchX4uL6DY53PzlRHzOZx_Tt2qxfH2fPi-qltjsK2-UjApV651G7UNjuI0RkIBVDCZqI2VTO-NJUYConKwVIDYE0CnfdXIi7k-9uzR8jyHv7XoYU3-YtAcdzcxK04F6OFFtGnJOIdrjX5d-LII96tt_ffkLF8RJcQ</recordid><startdate>20020101</startdate><enddate>20020101</enddate><creator>Man, Shing Hing</creator><creator>Smith, Patrick F.</creator><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20020101</creationdate><title>On chains of prime submodules</title><author>Man, Shing Hing ; Smith, Patrick F.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c257t-b743f414cba616be975cff012054fe7f673398a7b242e0f4a3840119200d4bdd3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2002</creationdate><topic>Chains</topic><topic>Domains</topic><topic>Mathematics</topic><topic>Modules</topic><topic>Rings (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Man, Shing Hing</creatorcontrib><creatorcontrib>Smith, Patrick F.</creatorcontrib><collection>CrossRef</collection><jtitle>Israel journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Man, Shing Hing</au><au>Smith, Patrick F.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On chains of prime submodules</atitle><jtitle>Israel journal of mathematics</jtitle><date>2002-01-01</date><risdate>2002</risdate><volume>127</volume><issue>1</issue><spage>131</spage><epage>155</epage><pages>131-155</pages><issn>0021-2172</issn><eissn>1565-8511</eissn><abstract>In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structure of the underlying ring. The dimension of finitely generated modules over a Dedekind domain is also studied. By examining the structure of prime submodules, a formula for the dimension of a free module of finite rank, over a Noetherian one-dimensional domain, is obtained.</abstract><cop>Heidelberg</cop><pub>Springer Nature B.V</pub><doi>10.1007/BF02784529</doi><tpages>25</tpages></addata></record>
fulltext fulltext
identifier ISSN: 0021-2172
ispartof Israel journal of mathematics, 2002-01, Vol.127 (1), p.131-155
issn 0021-2172
1565-8511
language eng
recordid cdi_proquest_journals_2786555462
source Springer Nature
subjects Chains
Domains
Mathematics
Modules
Rings (mathematics)
title On chains of prime submodules
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-26T07%3A50%3A50IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=On%20chains%20of%20prime%20submodules&rft.jtitle=Israel%20journal%20of%20mathematics&rft.au=Man,%20Shing%20Hing&rft.date=2002-01-01&rft.volume=127&rft.issue=1&rft.spage=131&rft.epage=155&rft.pages=131-155&rft.issn=0021-2172&rft.eissn=1565-8511&rft_id=info:doi/10.1007/BF02784529&rft_dat=%3Cproquest_cross%3E2786555462%3C/proquest_cross%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c257t-b743f414cba616be975cff012054fe7f673398a7b242e0f4a3840119200d4bdd3%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2786555462&rft_id=info:pmid/&rfr_iscdi=true