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Title:On Isolated Submodules 
Authors:
Roy McCasland
; Patrick F. Smith

Date:Aug 2006 
Publication Title:Communications in Algebra 
Publisher:Taylor & Francis 
Publication Type:Journal Article
Publication Status:Published

Volume No:34(8)
Page Nos:29772988

DOI:10.1080/00927870600639773

 Abstract:
 Let R be a ring with identity and let M be a unital left Rmodule. A proper submodule L of M is radical if L is an intersection of prime submodules of M. Moreover, a submodule L of M is isolated if, for each proper submodule N of L, there exists a prime submodule K of M such that N is contained in K but L is not contained in K. It is proved that every proper submodule of M is radical (and hence every submodule of M is isolated) if and only if N intersected with IM is equal to IN for every submodule N of M and every (left primitive) ideal I of R. In case, R/P is an Artinian ring for every left primitive ideal P of R it is proved that a finitely generated submodule N of a nonzero left Rmodule M is isolated if and only if PN equals N intersect PM for every left primitive ideal P of R. If R is a commutative ring then a finitely generated submodule N of a projective Rmodule M is isolated if and only if N is a direct summand of M.
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 @Article{EDIINFRR0591,
 author = {
Roy McCasland
and Patrick F. Smith
},
 title = {On Isolated Submodules},
 journal = {Communications in Algebra},
 publisher = {Taylor & Francis},
 year = 2006,
 month = {Aug},
 volume = {34(8)},
 pages = {29772988},
 doi = {10.1080/00927870600639773},
 url = {http://taylorandfrancis.metapress.com/openurl.asp?genre=article&eissn=15324125&volume=34&issue=8&spage=2977},
 }
