Editing Annihilator (ring theory) (section)

=== Over the integers ===
Over <math>\mathbb{Z}</math> any finitely generated module is completely classified as the direct sum of its [[free module|free]] part with its torsion part from the fundamental theorem of abelian groups. Then the annihilator of a finitely generated module is non-trivial only if it is entirely torsion. This is because
:<math>\text{Ann}_{\mathbb{Z}}(\mathbb{Z}^{\oplus k}) = \{ 0 \} = (0)</math>
since the only element killing each of the <math>\mathbb{Z}</math> is <math>0</math>. For example, the annihilator of <math>\mathbb{Z}/2 \oplus \mathbb{Z}/3</math> is
:<math>\text{Ann}_\mathbb{Z}(\mathbb{Z}/2 \oplus \mathbb{Z}/3) = (6) = (\text{lcm}(2,3)),</math>
the ideal generated by <math>(6)</math>. In fact the annihilator of a torsion module
:<math>M \cong \bigoplus_{i=1}^n (\mathbb{Z}/a_i)^{\oplus k_i}</math>
is [[isomorphic]] to the ideal generated by their [[least common multiple]], <math>(\operatorname{lcm}(a_1, \ldots, a_n))</math>. This shows the annihilators can be easily be classified over the integers.