How do we add elements in a quiver?

I know how to multiply this (by concatenation) but how do we add them, i.e how do we interpet ab - c ? I don't understand why ab- c is not an element of $R^{2}$.

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You should ask this on math.stackexchange.com – Mariano Suárez-Alvarez Oct 11 2011 at 15:48

Thanks, can you please explain why ab - c is not an element of $R^{2}$? – New Oct 11 2011 at 16:09
Is it simply because assuming it is true then ab is $R^{2}$ because it has length $2$ and since $R^{2}$ is an ideal then $R^{2}$ contains $ab-(ab-c)=c$ which has length $1$ so impossible? – New Oct 11 2011 at 16:11