(1,\bar{1}), (0,\bar{1}) is another basis, and the change of basis matrix \begin{bmatrix}1&0\\1&1\end{bmatrix} does not preserve the summand \mathbf{Z}. However, it does preserve the \mathbf{Z}/2 summand, as this is the torsion submodule (equivalently here, the 2-torsion elements).
