= Indecomposable projectives of S3 in characteristic three
{title2=$kS_3=e_+kS_3\oplus e_-kS_3$}
With $s=(12)$, the <idempotents> $e_\pm=(1\pm s)/2$ split the right regular <module> into two three-dimensional <projective modules>. Their tops are respectively the <trivial representation> and <sign representation>. The <nilpotent ideal> $J(kS_3)$ ensures that any nonzero <direct summand> has nonzero top, so the one-dimensional tops make these <projective modules> indecomposable. Their successive <radical series of a module> factors are trivial, sign, trivial and sign, trivial, sign. This decomposition concerns right <modules>; the <block of S3 in characteristic three> remains a single <algebra>.
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