= Block of S3 in characteristic three
{title2=$\operatorname{char}k=3$}
Over a <field> of <characteristic> three, the <group algebra> $A=kS_3$ has a single <block of an Artinian algebra>. Put $r=(123)$, $s=(12)$ and $a=r-1$. Then $a^3=0$, $sas=-a+a^2$, and $J(A)=aA$: the latter is a <nilpotent ideal>, with <semisimple ring> quotient $kC_2$. The <center of an associative algebra> is $k1\oplus ka^2\oplus ka^2s$, whose last two summands form a <square-zero ideal>. A <central idempotent> $d1+n$ satisfies $d^2=d$ and $(2d-1)n=0$, hence is zero or one. Thus no nontrivial central block decomposition exists.
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