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I Jacobson Radical and Right Annihilator Ideals ...

  1. Feb 7, 2017 #1
    I am reading Paul E. Bland's book, "Rings and Their Modules".

    I am focused on Section 6.1 The Jacobson Radical ... ...

    I need help with the proof of Proposition 6.1.7 ...


    Proposition 6.1.7 and its proof read as follows:


    ?temp_hash=57814fcc0e503006e7153e48b9211c10.png
    ?temp_hash=57814fcc0e503006e7153e48b9211c10.png



    In the above text from Bland ... in the proof of (1) we read the following:


    " ... ... we see that ##\text{ann}_r( R / \mathfrak{m} ) = \text{ann}_r(S)##. But ##a \in \text{ann}_r( R / \mathfrak{m} )## implies that ##a + \mathfrak{m} = ( 1 + \mathfrak{m} ) a = 0## , so ##a \in \mathfrak{m}##.

    So, it follows that ##\bigcap_\mathcal{S} \text{ann}_r(S) \subseteq J(R)## ... ... "



    Could someone please explain why it follows that ##\bigcap_\mathcal{S} \text{ann}_r(S) \subseteq J(R)## ... ... ?


    Hope someone can help ...

    Peter
     

    Attached Files:

  2. jcsd
  3. Feb 7, 2017 #2


    Just some thoughts on my own question ... ...


    Just some thoughts ...

    Since ##\text{ann}_r( R / \mathfrak{m} ) = \text{ann}_r(S)##

    we have ##a \in \text{ann}_r( R / \mathfrak{m} )## means ##a \in \text{ann}_r(S)## ...

    thus ##a \in \bigcap_\mathscr{S} \text{ann}_r(S)## ... ...

    But ... we also have that ##a \in \text{ann}_r( R / \mathfrak{m} )## implies that ##a \in \mathfrak{m}## ... ...

    But this means that ##a \in J(R)## ...

    Thus ##a \in \bigcap_\mathscr{S} \text{ann}_r(S) \Longrightarrow a \in J(R)## ... ...

    So ##\bigcap_\mathscr{S} \text{ann}_r(S) \subseteq J(R)## ...


    Is that correct?


    Peter
     
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