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Let M be R-module, and A,N submodules such that $A \subset N$ Let $f:M/A \to M/N$ be defined by $f(mA)=mN$ for $m\in M$.

I'm having trouble seeing why exactly this map is well-defined (It should be). This should be quite trivial... Please help. Thanks.

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You are mapping $m+A$ to $m+N$ and have to show that this is independent of the representing element of the equivalence class $mA$. So let $m=n\in M/A$. In other words, $m-n=a\in A$. You have to show that $f(m)-f(n)\in N$. So consider $$f(m)-f(n)=f(m-n)=f(a)=a\in A\subset N.$$ So your map is independent of the representing element and well-defined.

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  • $\begingroup$ I see. Thanks a bunch. $\endgroup$ – davidh Feb 20 at 6:31
  • $\begingroup$ You‘re welcome. $\endgroup$ – James Feb 20 at 6:37

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