Simple Modules and quotients of maximal modules, Bland Ex 13

[Math Amateur]

I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Chapter 1, Section 1.4 Modules ... ...

I need help with the proving a statement Bland makes in Example 13 ... ...

Example 13 reads as follows:

In the above text from Bland, we read the following:

" ... If ##N## is a maximal submodule of ##M##, then it follows that ##M/N## is a simple ##R##-module ... ... "I do not understand why this is true ... can anyone help with a formal proof of this statement ...
Hope someone can help ...

Peter

 

[fresh_42]

What are the submodules of ##M/N\,##? And what is the zero element in this factor module?

 

[Math Amateur]

What are the submodules of ##M/N\,##? And what is the zero element in this factor module?

Hi fresh_42 ...

I cannot answer you with confidence ... which is probably why I do not follow Bland Example 13 ... but ...

The elements of ##M/N## are the cosets ##\{ x + N \}_{ x \in M }## where ##x + N = \{ x + n \ | \ n \in N \}## ... ...

... BUT? ... what are the submodules of ##M/N## ... I am unsure ...

Zero element would be ##N = \{ 0 + N \}## ...

Can you help further ... ?

Peter

 

[fresh_42]

Hi fresh_42 ...

I cannot answer you with confidence ... which is probably why I do not follow Bland Example 13 ... but ...

The elements of ##M/N## are the cosets ##\{ x + N \}_{ x \in M }## where ##x + N = \{ x + n \ | \ n \in N \}## ... ...

... BUT? ... what are the submodules of ##M/N## ... I am unsure ...

Zero element would be ##N = \{ 0 + N \}## ...

Can you help further ... ?

Peter

Yes, exactly. But zero is in any submodule. So a submodule of ##M/N## as a set ##S := \{x + N \,\vert \, x \in \textrm{ something }\}## has to contain ##N##. Now ##N \subseteq S \subseteq M## is maximal, so ##S## is either equal to ##M## or equal to ##N##. But this means ##S/N = M/N## or ##S/N=N/N=\{0\}## which is the definition of a simple module: ##M/N## has no proper submodules ##S/N##.