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How do we add elements in a quiver?

Please see here:

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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closed as off-topic by Stefan Kohl, Andrey Rekalo, j.c., Carlo Beenakker, David White Nov 6 '13 at 14:13

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Stefan Kohl, Andrey Rekalo, j.c., Carlo Beenakker, David White
If this question can be reworded to fit the rules in the help center, please edit the question.

You should ask this on – Mariano Suárez-Alvarez Oct 11 '11 at 15:48

Just formally, think of non-commuting variables in a polynomial ring.

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Thanks, can you please explain why ab - c is not an element of $R^{2}$? – New Oct 11 '11 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 '11 at 16:11
Yes, that's right. – Julian Kuelshammer Oct 11 '11 at 16:18

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