# Simple Modules and quotients of maximal modules, Bland Ex 13

• I
• Math Amateur
In summary, Bland's statement in Example 13 holds true because if ##N## is a maximal submodule of ##M##, then ##M/N## is a simple ##R##-module.
Math Amateur
Gold Member
MHB
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 ... ...

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

#### Attachments

• Bland - Example 13 - Page 30, Ch. 1 ... ....png
14.7 KB · Views: 578
What are the submodules of ##M/N\,##? And what is the zero element in this factor module?

fresh_42 said:
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

Math Amateur said:
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##.

Math Amateur

## 1. What are simple modules?

A simple module is a module that has no proper submodules. In other words, it cannot be written as a direct sum of two non-zero submodules. Simple modules are important in the study of representation theory and ring theory.

## 2. How are simple modules related to maximal modules?

A maximal module is a module that is not a proper submodule of any other module. Simple modules are always maximal modules, but not all maximal modules are simple. This is because a maximal module can have a composition series that consists of simple modules.

## 3. What is the significance of Bland Ex 13 in relation to simple and maximal modules?

Bland Ex 13 is a theorem that states that every simple module is isomorphic to a quotient of a maximal module. This means that simple modules can be obtained by "modding out" a maximal module by one of its submodules. It also shows that maximal modules play a crucial role in understanding the structure of simple modules.

## 4. Can simple modules have non-trivial quotients?

Yes, simple modules can have non-trivial quotients. This means that a simple module can have a submodule that is not equal to the entire module. In fact, the only modules that do not have non-trivial quotients are the zero module and the simple module itself.

## 5. How do simple modules and quotients of maximal modules relate to the structure of a ring?

The study of simple modules and quotients of maximal modules is important in understanding the structure of a ring. In particular, simple modules help identify irreducible representations of the ring, while quotients of maximal modules provide information about the submodules and ideals of the ring. This can be useful in studying the ring's properties and determining its structure.

Replies
4
Views
1K
Replies
2
Views
1K
Replies
37
Views
4K
Replies
3
Views
1K
Replies
8
Views
2K
Replies
6
Views
2K
Replies
2
Views
1K
Replies
2
Views
1K
Replies
13
Views
2K
Replies
9
Views
2K