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Let $Q$ be a finite quiver. Then the following hold: (a) If $KQ$ is semisimple, then $|Q_1| = 0$.

If, moreover, $Q $ is connected, show that:

(b)$KQ$ is local only if $|Q_0| = 1$ and $|Q_1| = 0$,

Aimin Xu
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For (a) I guess it depends on the definition of semisimple. If you take Jacobson radical vanishes as the definition. Your claim is not true, since if you take the quiver with one vertex and a loop, you get the polynomial ring in one variable $K[X]$, whose Jacobson radical vanishes, but it has an arrow.

In Assem, Simson, Skowronski however, which seems to be the book you use (it is helpful if you include this data in your question), if I recall correctly, semisimple is only defined for finite-dimensional algebras. Then $KQ$ finite-dimensional implies that $Q$ has not oriented cycles. And in this case the ideal spanned by the arrows coincides with the Jacobson radical. Hence for the algebra to be semisimple there must not be arrows, hence $|Q_1|=0$.

For (b) if there are more than two vertices, then pick one of them, say $i$. Call the other $j$. Then $e_i$ is not invertible (since it is a zero divisor since $e_j\cdot e_i=0$). Similarly $1-e_i$ is not invertible. A contradiction to the property that for each local ring $x$ or $1-x$ is invertible. Thus $|Q_0|=1$. Now you are left with a polynomial ring (although with non-commuting variables). And you can use the excellent answers to your former question to conclude that there also must not be arrows.

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Julian Kuelshammer
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  • If $KQ$ is finite-dimensional, the proof is easy as you give. If it is infinitely dimensional, I can not prove (a) in this case. $KQ$ is semisimple in case $KQ$ is semisimple as a left $KQ$-module. – Aimin Xu Nov 03 '12 at 13:01
  • I think it is not true in the infinite-dimensional case, see e.g. http://math.stackexchange.com/questions/133737/semisimplicity-of-a-polynomial-ring, http://www.math.uni-bielefeld.de/~sek/select/rw1.pdf – Julian Kuelshammer Nov 03 '12 at 13:24
  • Dear Julian, can you explain why "Jacobson radical of $K[X]$ vanishes"? Isn't it the ideal generated by $X$? – Aaron Nov 05 '12 at 12:41
  • The Jacobson radical is the intersection of all maximal left ideals (since we are in the commutative setting, the left can be omitted). There are infinitely many of them, for infinite fields the $(X-\lambda)$ for $\lambda\in K$ are examples. For finite fields, see e.g. http://math.stackexchange.com/questions/70127/explicit-examples-of-infinitely-many-irreducible-polynomials-in-kx . Hence their intersection is $0$. – Julian Kuelshammer Nov 05 '12 at 12:52
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    If $D$ is a division ring, then the polynomial ring $R=D[x_1,x_2,\ldots x_n]$ is $J$-semisimple, that is, the radical $J(R)$ of $R$ is zero. For if $f\in J(R)$, then $1_{D}-f$ is a unit in $R$ . Since the only units in $R$ are the nonzero elements of $D$ , it follows that $f\in D$. Thus $J(R)$ is an ideal of $D$, whence $J(R)=0$ or $J(R)=D$ by the simplicity of $D$. Therefore $J(R)=0$ since $-1_{D}\notin J(R)$ ($1_{D}-1_{D}=0 $ is not a unit) – Aimin Xu Nov 08 '12 at 01:50