= Integral cohomology ring of the wedge of the real projective plane and a circle
{title2=$H^*(\mathbb{RP}^2\vee S^1;\mathbb Z)$}
The wedge $\mathbb{RP}^2\vee S^1$ has the same additive integral cohomology as the <Klein bottle>: $\mathbb Z$ in degree zero, $\mathbb Z$ in degree one, and $\mathbb Z/2$ in degree two. Products of positive-degree classes vanish because the degree-one class comes from the circle summand, so its <cohomology ring> is isomorphic to that of the Klein bottle.
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