To find annihilator of given module

Question

I want to find annihilator of $Z_{14}$ and $Z_4 × Z_6$ as Z module. So in first case element a from Z will be in annihilator of $Z_{14}$ if 14 divides a,2a,3a,....,13a. But from here I am not getting what values of a will work, may be it's simple but it's not getting click to me. I think if I got this , then second will also follow easily. Any help.

Answer 1

If $\{x_1,x_2,\dots,x_n\}$ is a set of generators of the $R$-module $M$, then $r\in R$ annihilates $M$ if and only if it annihilates all the generators.

The $\mathbb{Z}$-module $\mathbb{Z}_{14}$ has $1+14\mathbb{Z}$ as generator.

The $\mathbb{Z}$-module $\mathbb{Z}_4\times\mathbb{Z}_6$ has $(1+4\mathbb{Z},0+6\mathbb{Z})$ and $(0+4\mathbb{Z},1+6\mathbb{Z})$ as generators.

The integer $r$ annihilates $1+n\mathbb{Z}$ if and only if $r\in n\mathbb{Z}$.