On Isolated Submodules |
| |
Authors: | Roy L McCasland Patrick F Smith |
| |
Institution: | 1. School of Informatics, University of Edinburgh , Edinburgh, Scotland, UK rmccasla@inf.ed.ac.uk;3. Department of Mathematics , University of Glasgow , Glasgow, Scotland, UK |
| |
Abstract: | 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. |
| |
Keywords: | 16D10 13F10 16D60 16N60 16P20 |
|
|