# is every ideal of a ring an annihilator of an element of some module?

• November 30th 2011, 05:37 PM
ymar
is every ideal of a ring an annihilator of an element of some module?
Let $I$ be a right ideal of a ring $R.$ Is there always a right $R-$module $M$ and $m\in M$ such that $I=\mathrm{ann}(m)?$
• November 30th 2011, 07:05 PM
NonCommAlg
Re: is every ideal of a ring an annihilator of an element of some module?
Quote:

Originally Posted by ymar
Let $I$ be a right ideal of a ring $R.$ Is there always a right $R-$module $M$ and $m\in M$ such that $I=\mathrm{ann}(m)?$

yes, $M=R/I$ and $m = 1+I.$
• November 30th 2011, 07:10 PM
Drexel28
Re: is every ideal of a ring an annihilator of an element of some module?
Quote:

Originally Posted by ymar
Let $I$ be a right ideal of a ring $R.$ Is there always a right $R-$module $M$ and $m\in M$ such that $I=\mathrm{ann}(m)?$

I assume that our rings are unital, no? Note then that $R/I$ has a natural $R$-module structure. If $x\in I$ then we see that $(r+I)x=rx+I=0$ since $rx\in I$ for all $r\in R$ and so $x\in\text{ann}(R/I)$. Conversely, if $x\in\text{ann}(R/I)$ then $0=(1+I)x=x+I$, so that $x\in I$, etc.

EDIT: Misread to try and prove that $\text{ann}(M)=I$, but NonCommAlg read it correctly. Same idea though.