Jacobson Radical and Right Annihilator Ideals ....

[Math Amateur]

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

 

[Math Amateur]

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

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