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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:

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

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:

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