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Let $A$ be a $k$-algebra. The following are equivalent: (i) $A$ is indecomposable as a $k$-algebra. (ii) $A$ is indecomposable as an $A-A$-bimodule. (iii) The idempotent $1_A$ is primitive in $Z(A)$.

Then the author says later that $M_n(k)$ is an indecomposable algebra. Why? Apprarently, the identity matrix is not primitive(can be written as the sum of two orthogonal idempotents)

Any help would be appreciated!

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    $\begingroup$ Is $Z(A)$ the center? If so, what is $Z(M_n(k))$? $\endgroup$
    – Aphelli
    Aug 29, 2020 at 10:23
  • $\begingroup$ I see. thank you for the hints! $\endgroup$
    – scsnm
    Aug 29, 2020 at 10:29

1 Answer 1

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The identity matrix isn’t primitive in $M_n(k)$, but it is primitive in $Z(M_n(k))=k$, which is what the third thing says.

Any simple $k$-algebra $A$ has to be indecomposable, of course, because a nontrivial central idempotent $e$ would create a nontrivial ideal $eA$

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