 Heisenberg group over a commutative ring

For a commutative ring $R$ with an identity element, use coordinatesThese matrices form a group under multiplication, withThese formulas require no division, so they remain valid over rings with zero divisors. The center of a group is $\{A(0,y,0):y\in R\}$: the commutator has only central coordinate $x{z}^{\prime }-x^{\prime }z$, and testing $x^{\prime }=0,z^{\prime }=1$ and $x^{\prime }=1,z^{\prime }=0$ forces $x=z=0$. For finite $R$, its group order is $|R{|}^3$. This includes real, integer and modular versions of the Heisenberg group; the ring $\mathbb{Z}/4\mathbb{Z}$ must not be confused with the finite field of four elements.