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It is known that a finitely generated torsion module $M$ over a principal ideal domain $R$ can be decomposed into a direct sum of primary modules, $$ M=M_{p_1}\oplus\cdots\oplus M_{p_n}. $$ Futhermore, the primary submodules $M_{p_i}$ can be decomposed as a direct sum of cyclic submodules, but this decomposition is not unique.

Is there a standard example exhibiting why the cyclic decomposition of a primary module is not unique?

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Sure, consider an abelian group of order $4$, written additively, which is the sum $$ \langle a \rangle \oplus \langle b \rangle, $$ with $a$ and $b$ of order $2$. Then you also have $$ \langle a+b \rangle \oplus \langle b \rangle. $$ So you see that this is just the fact that in general a vector space (over $\mathbf{Z}_{2}$, in this case) does not have a unique basis.

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