Abstract Algebra

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Annihilator of a Module: Example of Product of Abelian Groups

ABSALG-4OD6QR

Let $R$ be a ring and $M$ a left $R$-module. Let $N$ be an $R$-submodule of $M$.

Define the annihilator ${\rm Ann}(N)$ of $N$ in $R$ by:

$${\rm Ann}(N):=\{r\in R\mid rn=0,\;{\rm for\; all}\; n\in N\}$$

Let $M$ be the $\mathbb {Z}$-module $M=\mathbb {Z}/30\mathbb{Z}\times\mathbb{Z}/42\mathbb{Z}$.

Which of the following equals ${\rm Ann}(M)$?

A

$1260\mathbb{Z}$

B

$6\mathbb{Z}$

C

$210\mathbb{Z}$

D

$30\mathbb{Z}\times 42\mathbb{Z}$