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From Dummit Foot, 10.1 ex.11b)

Let M be the abelian group $\mathbb{Z}/\mathbb{24Z}\times \mathbb{Z}/\mathbb{15Z}\times\mathbb{Z}/\mathbb{50Z}$. Let $I=2\mathbb{Z}$ and describe the annihilator I in M as a direct product of cyclic groups.

For reference, If I is a right ideal of R, the annihilator of I in M is $\{m\in M| am=0 ~\forall a\in I \}$

Thanks.

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    @BenjaminLim: Abelian groups are $\mathbb{Z}$-modules. Multiplication by $n\cdot m=m+m+\ldots m$ ($n$ times).2012-02-26

3 Answers 3

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The elements of $2\mathbb{Z}$ are even integers i.e. of the form $2m$ for $m\in\mathbb{Z}$. Write an arbitrary $x\in\mathrm{Ann}(I)$ in component form as $(a,b,c)\in M$. Since $2x=0$ implies $(2m)x=0$ for all $m\in\mathbb{Z}$, it suffices to check that $2$ annihilates $x$ in $M$. This can be written as the following congruence system:

$2a\equiv0\quad (24)$ $2b\equiv 0\quad (15)$ $2c\equiv0\quad (50)$

In order, this implies $12|a$, $15|b$ and $25|c$ - this is elementary number theory.

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The annihilator of $I$ in $M$ is $\{(x,0,y)\in M:12|x, 25|y\}\cong \mathbb{Z}/2\oplus\mathbb{Z}/2$. Try acting on such an element by an element of $I$ (for example, multiplying each coordinate by 2), and do the same with an element not in the annihilator, and it should become clear how I came up with the answer.

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If you first prove some basic facts about the notion of "the annihilator of an ideal in a module", it becomes a simple computation: $\begin{align*} & \mathrm{Ann}_{2\mathbb{Z}}(\mathbb{Z}/\mathbb{24Z}\times \mathbb{Z}/\mathbb{15Z}\times\mathbb{Z}/\mathbb{50Z}) \\ &= \mathrm{Ann}_{2\mathbb{Z}}(\mathbb{Z}/\mathbb{24Z})\times \mathrm{Ann}_{2\mathbb{Z}}(\mathbb{Z}/\mathbb{15Z})\times\mathrm{Ann}_{2\mathbb{Z}}(\mathbb{Z}/\mathbb{50Z}) \\ &= (\{0,12\} \subseteq \mathbb{Z}/\mathbb{24Z}) \times (\{0\} \subseteq \mathbb{Z}/\mathbb{15Z}) \times (\{0,25\} \subseteq \mathbb{Z}/\mathbb{50Z}) \\ & \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \end{align*}$

The idea is to look at the above computation, convince yourself each step makes sense, then formalize the theorem used in each step as a precise statement of mathematics and prove it.