On Isolated Submodules

Let R be a ring with identity and let M be a unital left R-module. 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 ⊆ K but L ⊈ K. It...

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Bibliographic Details
Published in:Communications in algebra 2006-08, Vol.34 (8), p.2977-2988
Main Authors: McCasland, Roy L., Smith, Patrick F.
Format: Article
Language:English
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Summary:Let R be a ring with identity and let M be a unital left R-module. 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 ⊆ K but L ⊈ 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 ∩ IM = 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 R-module M is isolated if and only if PN = N ∩ PM for every left primitive ideal P of R. If R is a commutative ring, then a finitely generated submodule N of a projective R-module M is isolated if and only if N is a direct summand of M.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927870600639773