= Solution
{c}
Define the <group homomorphism>
$$
\theta:\mathbb Z\longrightarrow\mathbb Z_d\oplus\mathbb Z_h,\qquad
t\longmapsto(t+d\mathbb Z,t+h\mathbb Z).
$$
By part (b)(i), its <kernel of a group homomorphism> is
$$
\ker\theta=d\mathbb Z\cap h\mathbb Z=dh\mathbb Z.
$$
It is <surjective>. Indeed, choose $r,s\in\mathbb Z$ with $rd+sh=1$ by <Bézout's identity>. For any prescribed classes $u+d\mathbb Z$ and $v+h\mathbb Z$, the <integer>
$$
t=ush+vrd
$$
satisfies $t\equiv u\pmod d$ and $t\equiv v\pmod h$. The <first isomorphism theorem for groups> therefore gives
$$
\boxed{\mathbb Z_{dh}\xrightarrow{\ \cong\ }\mathbb Z_d\oplus\mathbb Z_h,\qquad
t+dh\mathbb Z\longmapsto(t+d\mathbb Z,t+h\mathbb Z)}.
$$
This proves the required isomorphism and is the <Chinese remainder theorem> for <coprime integers>. It also shows that $(1+d\mathbb Z,1+h\mathbb Z)$ generates the <direct sum of modules> when these additive <groups> are viewed as $\mathbb Z$-<modules>.
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