= Integral Heisenberg group algebra as a skew Laurent ring
{title2=$kH\cong k[y^{\pm1},z^{\pm1}][x^{\pm1};\sigma],\quad\sigma(y)=zy$}
For the <Integer Heisenberg group>, choose $x=I+E_{12}$, $y=I+E_{23}$ and central $z=I+E_{13}$. Every element has a unique form $y^cz^bx^a$, and $xy=zyx$. The <group algebra> is therefore the displayed <skew Laurent polynomial ring>, with $\sigma(z)=z$. The base is a commutative <Laurent polynomial ring> and an <integral domain>; the <skew Hilbert basis theorem> gives both-sided Noetherianity and the <leading coefficient> of a product proves that there are no zero divisors.
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