$M_n(k)$ is an indecomposable algebra 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!

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 $eA$
