Editing Annihilator (ring theory) (section)

== Examples ==

=== 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.

=== Over a commutative ring ''R'' ===
There is a similar computation that can be done for any [[finitely presented module]] over a commutative ring <math>R</math>. The definition of finite presentedness of <math>M</math> implies there exists an exact sequence, called a presentation, given by
:<math>R^{\oplus l} \xrightarrow{\phi} R^{\oplus k} \to M \to 0</math>
where <math>\phi</math> is in <math>\text{Mat}_{k,l}(R)</math>. Writing <math>\phi</math> explicitly as a [[matrix (mathematics)|matrix]] gives it as
:<math>\phi = \begin{bmatrix}
\phi_{1,1} & \cdots & \phi_{1,l} \\
\vdots &  & \vdots \\
\phi_{k,1} & \cdots & \phi_{k,l}
\end{bmatrix};</math>
hence <math>M</math> has the direct sum decomposition
:<math>M = \bigoplus_{i=1}^k \frac{R}{(\phi_{i,1}(1), \ldots, \phi_{i,l}(1))}</math>
If each of these ideals is written as
:<math>I_i = (\phi_{i,1}(1), \ldots, \phi_{i,l}(1))</math>
then the ideal <math>I</math> given by
:<math>V(I) = \bigcup^{k}_{i=1}V(I_i)</math>
presents the annihilator.

=== Over ''k''[''x'',''y''] ===
Over the commutative ring <math>k[x,y]</math> for a [[field (mathematics)|field]] <math>k</math>, the annihilator of the module
:<math>M = \frac{k[x,y]}{(x^2 - y)} \oplus \frac{k[x,y]}{(y - 3)}</math>
is given by the ideal
:<math>\text{Ann}_{k[x,y]}(M) = ((x^2 - y)(y - 3)).</math>