Chinese remainder theorem
Decomposition of surjections of finite abelian groups

Given a surjection Z / n → Z / m {\displaystyle \mathbb {Z} /n\to \mathbb {Z} /m} of finite abelian groups, we can use the Chinese remainder theorem to give a complete description of any such map. First of all, the theorem gives isomorphisms

where { p m 1, …, p m j } ⊆ { p n 1, …, p n i } {\displaystyle \{p_{m_{1}},\ldots,p_{m_{j}}\}\subseteq \{p_{n_{1}},\ldots,p_{n_{i}}\}}. In addition, for any induced map

from the original surjection, we have a k ≥ b l {\displaystyle a_{k}\geq b_{l}} and p n k = p m l, {\displaystyle p_{n_{k}}=p_{m_{l}},} since for a pair of primes p, q {\displaystyle p,q}, the only non-zero surjections

can be defined if p = q {\displaystyle p=q} and a ≥ b {\displaystyle a\geq b}.

These observations are pivotal for constructing the ring of profinite integers, which is given as an inverse limit of all such maps.
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