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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 Corollary 6.1.3 ...Corollary 6.1.3 (including the preceding Proposition) reads as follows:

My questions are as follows:

In the proof of Corollary 6.1.3 above we read:

"... ... Since ##R## is generated by ##1, J(R) \neq R##. ... ...My question is as follows: why, given that ##R## is generated by ##1##, is it true that ##J(R) \neq R## ... ... ?

Bland seems to argue that if we accept that ##J(R) \neq R##, then the Corollary is proved ... ... that is that

##J(R) \neq R \Longrightarrow \text{ Rad}(M) \neq M## ... ...

But ... why would this be true ...?Hope someone can help ... ...Peter

===========================================================================In order to give forum readers the notations, definitions and context of the above post, I am providing the first two pages of Chapter 6 of Bland ... ... as follows ... ... :

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

I need help with the proof of Corollary 6.1.3 ...Corollary 6.1.3 (including the preceding Proposition) reads as follows:

My questions are as follows:

**Question 1**In the proof of Corollary 6.1.3 above we read:

"... ... Since ##R## is generated by ##1, J(R) \neq R##. ... ...My question is as follows: why, given that ##R## is generated by ##1##, is it true that ##J(R) \neq R## ... ... ?

**Question 2**Bland seems to argue that if we accept that ##J(R) \neq R##, then the Corollary is proved ... ... that is that

##J(R) \neq R \Longrightarrow \text{ Rad}(M) \neq M## ... ...

But ... why would this be true ...?Hope someone can help ... ...Peter

===========================================================================In order to give forum readers the notations, definitions and context of the above post, I am providing the first two pages of Chapter 6 of Bland ... ... as follows ... ... :