= Semigroup algebra
{title2=$k[D]$}
For an additive <semigroup> $D$ with identity zero, the semigroup algebra $k[D]$ has a $k$-basis of formal symbols $t^q$, $q\in D$, and multiplication $t^qt^{q'}=t^{q+q'}$. For $D=\mathbb Z[1/2]_{\ge0}$ it is the <integral domain> $\bigcup_{h\ge0}k[t^{1/2^h}]$. Its positive-exponent <ideal> is a nonzero <idempotent ideal>, and the strictly ascending chain $(t)\subsetneq(t^{1/2})\subsetneq\cdots$ shows that the algebra is not a <Noetherian ring>.
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