Proove an isomorphism

**Carl** · November 18th 2010, 07:23 AM

Hi there

I'm a little stuck in this one and could use a hand:
I need to show this isomorphism:
$(N_1+N_2)/N_2 \simeq N_1/(N_1 \cap N_2)$

Well, I make an exact sequence:

$0\rightarrow N_1 \underrightarrow{\iota} (N_1 +N_2) \underrightarrow{\kappa} (N_1+N_2)/N_2 \rightarrow 0$

where $\iota$ is the inclusion and $\kappa$ the canonical mapping.
And then I find the Kernel of $\kappa \circ \iota$ but how do I get on from here? As far as I have understood I must find that the Kernel is exactly $(N_1 \cap N_2)$ .

**topspin1617** · November 18th 2010, 07:29 AM

Right idea, but you need to include one more detail:

We know the natural map $\kappa$ is surjective. But to use the First Isomorphism Theorm, we need to know that the map $\kappa \circ \iota$ is surjective. Why is this? (Consider what an arbitrary element of $N_1+N_2/N_2$ would look like.)

Showing that the kernel of $\kappa \circ \iota$ is equal to $N_1\cap N_2$ should be fairly easy. What would it mean for an element to be in the kernel of this map?

**Carl** · November 19th 2010, 02:32 AM

Ok, I think I got it now, thanks.

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