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Essential: Let R be a ring. A submodule
N of an R-module M is said to be essential in
M if whenever L≠0 is a submodule of M, then
N∩L≠0.
The idea behind this photograph is to portray the "expansiveness" of an essential submodule. To use a linguistic metaphor, an essential submodule N of a module M is a submodule that has its fingers in the pots of every one of M's submodules: if L is any nontrivial submodule of M, then N and L must have a nontrivial intersection. N need not contain many elements, however; its significance in M is due to its "density" in M, so to speak, rather than to its size. This photograph attempts to convey this density: the tree (our submodule) fills the picture (our module), leaving no room in which other submodules can play without its leaves getting in the way..
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