If $\mathbb{Z_{mn}} \cong \mathbb{Z_m} \oplus \mathbb{Z_n}$ is true then by the definition of direct sum, must $\mathbb{Z_{mn}} = \mathbb{Z_m} + \mathbb{Z_n}$ hold, and for a special case of $n=2$ and $m=3$ it means that there exist integers $a, b$, and also $N_1 \in \mathbb{Z_2}$ and $M_1 \in \mathbb{Z_3}$ such that ${\{5,11,17,23, \dots}\} = a N_1 + b M_1$. But I can't find $a, b, N_1$ and $M_1$. So how $\mathbb{Z_{mn}} \cong \mathbb{Z_m} \oplus \mathbb{Z_n}$ is consistent with its definition?
Checking $\mathbb{Z_{mn}} \cong \mathbb{Z_m} \oplus \mathbb{Z_n}$ by the definition of direct sum
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abstract-algebra
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0When you say $\Bbb Z_{mn}=\Bbb Z_m+\Bbb Z_n$, do you really mean $mn\Bbb Z=m\Bbb Z\cap n\Bbb Z$? – 2017-02-09
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1Why do you think $\mathbb{Z}_{mn} \cong \mathbb{Z}_m\oplus\mathbb{Z}_n$ means $\mathbb{Z}_{mn} =\mathbb{Z}_m + \mathbb{Z}_n$? Is this what you think the definition literally is, or is this something you think follows from the definition? If the former, you should recheck the definition. If the latter, you should provide the definition of direct sum (i.e. $\oplus$) that you are using. – 2017-02-09
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0@DerekElkins. by def. of direct sum $M \cong M_1 \oplus M_2$ if $M = M_1 + M_2$ and $0 = M_1 \cap M_2$ – 2017-02-09
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1@Edi You should be careful with your notations, since for $M_1 +M_2$ to be well defined you need to first define what is the addition between the two groups. In your notation $\mathbb{Z}_n,\mathbb{Z}_m$ are two groups without any addition defined on both of them together, unless you identify them as subgroups of $\mathbb{Z}_{nm}$. – 2017-02-09