Degree-one cup-square obstruction to ring isomorphism ($z\in H^1,z^2\ne 0$)

The property that every degree-one element has square zero is invariant under a graded ring isomorphism. Over ${\mathbb{F}}_2$ it distinguishes the cohomology ring of a torus from the mod-two cohomology ring of an inversion mapping torus of dimension at least two.