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A basis algebra is an algebra $A$ such that $e_i A \not\simeq e_j A$ for any $i \neq j$, where $e_1, \ldots, e_n$ is a complete set of primitive orthogonal idempotents. A local algebra is an algebra with only one maximal right ideal. How to show that a local finite dimensional algebra is basic from definition? Thank you very much.

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  • $\begingroup$ I assume that the algebra is unital. Does a complete set of primitive orthogonal idempotents always exist (in finite dimensional unital algebras)? $\endgroup$ – Berci Mar 21 '13 at 2:30
  • $\begingroup$ @Berci, thank you very much. Yes, the algebra $A$ is unital. $\endgroup$ – LJR Mar 21 '13 at 2:36
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    $\begingroup$ @Berci, it seems that if $A$ is local, then $A$ has only two idempotents $0, 1$ and $A = 1A$. So by definition, $A$ is basic. Is this true? $\endgroup$ – LJR Mar 21 '13 at 2:39
  • $\begingroup$ do you mean basic algebra, not basis algebra? $\endgroup$ – Aaron Mar 21 '13 at 14:03
  • $\begingroup$ @LJR Your comment answers your question. You can post it and accept it, so that it gets removed from the list of unanswered questions. $\endgroup$ – Julian Kuelshammer Jun 8 '13 at 10:01
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If $A$ is local, then $A$ has only two idempotents $0,1$ and $A=1A$. So by definition, $A$ is basic.

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