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Math Help - is every ideal of a ring an annihilator of an element of some module?

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    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)?
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    Re: is every ideal of a ring an annihilator of an element of some module?

    Quote Originally Posted by ymar View Post
    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.
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    MHF Contributor Drexel28's Avatar
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    Re: is every ideal of a ring an annihilator of an element of some module?

    Quote Originally Posted by ymar View Post
    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.
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