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 $S/N$.