Uniqueness of decomposition of $Z_n $ as a $Z $-module

Question

Is it true that if $n=p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_m^{\alpha_m}$ ,then we have

$Z_n = Z_{p_1^{\alpha _1}}×\ldots Z_{p_m^{\alpha_m}}$

As $Z $-modules?

And this decomposition is unique up to a permutation?

How can I prove these?

Thanks

Answer 1

The decomposition is correct and is unique up to a permutation.

It follows from the Chinese remainder theorem, which is the fundamental theorem of arithmetic in an algebraic guise.

It is also an instance of the structure theorem for finitely generated modules over a principal ideal domain, namely, primary decomposition.