Reduced $K$-groups are ideals of the standard $K$-groups.  $\tilde K(X) \subset K(X)$ is the ideal of virtual-dimension-zero elements.

In particular, the reduced K-theory $\tilde K(S^2)$ is not $\mathbb{Z}[H]/(H-1)^2$, but rather the ideal of this generated by $(H-1)$.  In particular, any element in this group does square to zero.

Additionally, the "exterior product" isomorphism $f: \tilde K(X) \to \tilde K(X \wedge S^2)$, which is an isomorphism, is not a ring map: it takes an element $x$ to the exterior product $x \wedge (H-1)$.  Instead, it satisfies $f(x) f(y) = (H-1) f(xy) = 0$.  This is because the suspension is covered by two contractible open subsets, and so all products must vanish.