I am slightly confused by some statements in Hatcher's Vector Bundle book (page 60). To start with (and I am happy with) the natural ring homomorphism $K(S^2) \simeq \mathbb{Z}[H]/(H-1)^2$
I am trying to understand this statement
The external product $K(S^{2k}) \otimes K(X) \to K(S^{2k} \times X)$ is an isomorphism. This follows from (2) by the same reasoning which showed the equivalence of the reduced and unreduced forms of Bott periodicity. Since external product is a ring homomorphism, the isomorphism $K(S^{2k} \smash X) \simeq K(S^{2k}) \otimes K(X)$ is a ring isomorphism.
(I am okay up to here)
For example, since $K(S^{2k})$ can be described as the quotient ring $\mathbb{Z}[\alpha]/(\alpha^2 )$, we ...
(where he defines $\alpha$ as the pullback of the generator of of $\tilde{K}(S^{2k})$ under the projection $S^{2k} \times S^{2 \ell} \to S^{2k}$).
I don't see why it is not $\mathbb{Z}[\alpha]/((\alpha-1)^2 )$?