= Head-socle identity for a symmetric algebra
{title2=$\operatorname{Soc}P\cong P/\operatorname{Rad}P$}
For a finite-dimensional <symmetric algebra (Frobenius algebra)> over any <field>, an indecomposable <projective module> $P=Ae$ has isomorphic simple <head of a module> and <socle>. A symmetrizing functional pairs $eA$ with $Ae$ nondegenerately, identifying $Ae$ with the left <dual module> $D(eA)$. Taking its <socle> gives $D(eA/eAJ(A))$. This dual of the right simple top and the left simple top $Ae/J(A)e$ belong to the same matrix block of the <semisimple algebra> $A/J(A)$, so they are isomorphic. The result does not require an algebraically closed ground <field>.
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