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Let $R$ be a finite dimensional $k$-algebra given by a quiver. Then what is the quiver $R[x]/(x^2)$? For example, an algebra is given by the following quiver $$1\stackrel{\alpha}\rightarrow2$$ $$1\stackrel{\alpha, \beta}\rightleftharpoons2 \text{ with }\alpha\beta=0$$ $$1\circlearrowleft \text{with } rad^2=0$$ then what is the quiver of $R[x]/(x^2)$?

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    $\begingroup$ (Did you actually check how your question compiles?) $\endgroup$
    – Jules Lamers
    Commented Aug 11 at 8:55
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    $\begingroup$ I can't read your diagrams, but just add an extra loop marked $x$ at each vertex, and relations saying these loops square to zero and commute with all arrows that were already there. $\endgroup$
    – Dave Benson
    Commented Aug 11 at 11:54
  • $\begingroup$ Thank you! I edit it again. $\endgroup$
    – hgc
    Commented Aug 12 at 1:49

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