(This is exercise 10 of chapter 2 of Atiyah and Macdonald.)

The exercise starts by asking me to prove that if $A^n\cong A^m$ then $n=m$ for any nonzero ring A. I managed to do that (by tensoring with the residue field of a maximal ideal to get some nice vector spaces) - however, I can't seem to reconcile why the following is not a counterexample.

Let $A$ denote $\mathbb Z^{\aleph_0}$, the ring of sequences of integers. Then the map $\phi:A^2\to A$ given by interleaving sequences is an isomorphism. (If two sequences interleaved are zero, then both sequences were zero. Hence, $\ker\phi=0$ Any sequence can be decomposed into its even and odd subsequences, in which case applying $\phi$ will reconstruct the original. Therefore $\mathrm{im}\ \phi=A$.) Nonetheless, 2>1. What am I missing?

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