Let $R$ be an arbitrary ring with $1\neq 0$ and $_RM$ a left $R$-module. Is it true to say that : Every proper submodule of a module $M$ is contained in a maximal submodule?
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“Every nonzero module has a maximal submodule” is quite strong a property for the ring. See this article by Hamsher for the commutative case – egreg Aug 09 '18 at 10:53
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Not all modules have maximal submodules at all, much less maximal submodules that contain any given submodule.
As an example, the (left) $\Bbb Z$-module $\Bbb Q$ does not have maximal submodles.
Arthur
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1@KennyLau It's done here, for instance. That post is using the term "group", but that's the same as $\Bbb Z$-module for our intents and purposes. – Arthur Aug 09 '18 at 10:11