What is meant by $\text{Ann}(M)$ and $(0:_M a)$ in this example?

Question

In the answer here Is every Artinian module over an Artinian ring finitely generated? can you please tell me what is explicitly $Ann(M)$ and $(0:_Ma)$

thanks in advance

score 1 · Answer 1 · answered Feb 18 '18 at 21:35

1

For an $R$-module $M$

$\text{Ann}_R(M)$ is the set of $r\in R$ such that $r\cdot m=0$ for every $m\in M$. This is called the annihilator of the $R$-module $M$.
$(I:J)$ usually denotes the colon ideal, which is the set of $r\in R$ such that $rJ\subseteq I$. In your context I think $(0:_M a)$ means the set of $m\in M$ such that $am=0$.

answered Feb 18 '18 at 21:35

and why $(0:_Ma)\not=M$ in the answer ? I did not understand that ? – basas Feb 18 '18 at 21:53
@basas If $(0:_M a)=M$, then this would mean that $aM=0$, which is the same as saying that $a\in\text{Ann}(M)$. But they assumed that $a\not\in\text{Ann}(M)$ – cansomeonehelpmeout Feb 18 '18 at 21:56

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