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What does ''$\le$'' mean here?

Do you know the meaning of $\le$ in the second last line in the text below?

  1. The sequence $0\to N \to M \to M/N \to 0$ is exact, so by Problem 5, the sequence $0 \to N_S\to M_S \to (M,/N)_S \to 0$ is exact. (If f is one of the maps of the first sequence,the corresponding map in the second sequence is $S^{-1}f$.) It fellows from the definition of localization of a module that $N_S\le M_S$, and by exactness of the second sequence we have $(M/N)_S\cong M_S/N_S$.
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2 Answers 2

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It means submodule. In fact when there is an inclusion map $K \to L$, then we consider $K$ as submodule of $L$.
Here we have $0\to N\to M$ which gives $0\to N_S\to M_S$. So one can consider $N_S$ as submodule of $M_S$.

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It means that $N_S \subset M_S$ where $ N_S$ and $M_S$ are both modules.

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  • $\begingroup$ And the module structure on $N_S$ is the restriction of the module structure of $M_S$. $\endgroup$ Apr 16, 2015 at 21:47

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