Integral cohomology ring of the Klein bottle ($H^{\ast }(K;\mathbb{Z})$)

The integral cohomology groups of the Klein bottle are $H^0(K;\mathbb{Z})=\mathbb{Z}$, $H^1(K;\mathbb{Z})=\mathbb{Z}$, $H^2(K;\mathbb{Z})=\mathbb{Z}/2$, and zero in higher degrees. The degree-one generator is pulled back from the base circle of the circle-bundle presentation, so its square is zero; hence every product of positive-degree classes vanishes.

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.