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 $\mathbb{Z}[H-1]/(H-1)^2$, but rather the ideal of this generated by $(H-1)$.  In particular, any element in this group does square to zero.