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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