= Cocenter of a finite-dimensional algebra
{title2=$C(A)=A/[A,A]$}
The <cocenter of a finite-dimensional algebra> is the quotient by the vector subspace spanned by additive commutators $ab-ba$; this subspace need not be an <ideal>. For a <group algebra>, its basis is indexed by <conjugacy classes>: every commutator of basis elements is a difference of conjugates, and every difference of conjugates is a commutator. For $M_n(k)$, off-diagonal matrix units and $E_{ii}-E_{jj}$ show that the quotient is one-dimensional, identified by the <trace>.
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