= Top of an indecomposable projective module
{title2=$P/PJ(A)$}
For a nonzero finitely generated <indecomposable module> $P$ that is a <projective module> over a <right Artinian ring> $A$, its top $P/PJ(A)$ is a <simple module>. The <Jacobson radical> $J(A)$ is a <nilpotent ideal>, so the top is nonzero; the quotient is a <semisimple module> over the <semisimple ring> $A/J(A)$. If it decomposed, a nontrivial <idempotent> of its <endomorphism ring> would lift by projectivity to an <endomorphism> of $P$. The <Hopkins-Levitzki theorem> gives finite <composition length>, and the <Fitting lemma> makes the lift either invertible or nilpotent. Its induced map on the top would have the same property, impossible for that nontrivial <idempotent>.
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