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Abstract Algebra

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Annihilator of Element or Submodule: Left, Right,Two-Sided Ideal?

ABSALG-E1HZHL

Let $R$ be a ring which is not necessarily commutative.

Let $M$ be a left $R$-module, let $N$ be a left $R$-submodule of $M$, and let $m$ be an element of $M$.​

Let ${\rm Ann}(N)=\{r\in R\mid rn=0, \;{\rm for\;all}\; n\in N\}$ and let ${\rm Ann}(m)=\{r\in R\mid rm=0\}$.

${\rm Ann}(N)$, $N\not=M$, is always a

ideal of $R$, and ${\rm Ann}(m)$, $m\not=0$, is always a

ideal of $R$. It is always true that ${\rm Ann}(0)$ ($m=0$) is a

ideal of $R$. Finally ${\rm Ann}(M)$ is always a

ideal of $R$.