Exercise 111.11.1. Compute all the Ext groups \mathop{\mathrm{Ext}}\nolimits ^ i(M, N) of the given modules in the category of \mathbf{Z}-modules (also known as the category of abelian groups).
M = \mathbf{Z} and N = \mathbf{Z},
M = \mathbf{Z}/4\mathbf{Z} and N = \mathbf{Z}/8\mathbf{Z},
M = \mathbf{Q} and N = \mathbf{Z}/2\mathbf{Z}, and
M = \mathbf{Z}/2\mathbf{Z} and N = \mathbf{Q}/\mathbf{Z}.
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