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Abstract Algebra

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Moderate

Annihilators

ABSALG-NYDXNZ

Let $R$ be a commutative ring with unity, let $M$ be an $R$-module and $N$ be a submodule of $M$.
Define $Ann_R(N)=\{r\in R:rn=0\; \text{for all n}\in N\}$
$Ann_R(N)$ is called the annihilator of $N$.

Which of the following statements is NOT true in general?

Select ALL that apply.

A

$Ann_R(N)$ is an ideal of $R$

B

If $N_1\subseteq N_2$, then $Ann_R(N_1)\subseteq Ann_R(N_2)$

C

There exists a submodule $N$ with $Ann_R(N)=\{0\}$

D

$Ann_R(N_1+N_2)=Ann_R(N_1)+Ann_R(N_2)$

E

$R$ is a domain iff $Ann_R(R)=\{0\}$