Direct Sums of Noetherian Modules .... Bland Proposition 4.2.7 .... ....

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In summary, the proposition states that each M_i is isomorphic to a submodule of the direct sum of all M_i. This is proven by defining a natural embedding of M_i into the direct sum, which is shown to be an injective R-module homomorphism. This proves that each M_i is isomorphic to a submodule of the direct sum.
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I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.7 ... ...

Proposition 4.2.7 reads as follows:https://www.physicsforums.com/attachments/8208In the above proof by Paul Bland we read the following:

" ... ... and since each \(\displaystyle M_i\) is isomorphic to a submodule of \(\displaystyle \bigoplus_{ i = 1 }^n M_i\) ... ... "Can someone please explain to me how/why each \(\displaystyle M_i\) is isomorphic to a submodule of \(\displaystyle \bigoplus_{ i = 1 }^n M_i\) ... ... ?

Do we know what the submodule is ... ?
Help will be appreciated ...

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

I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.7 ... ...

Proposition 4.2.7 reads as follows:In the above proof by Paul Bland we read the following:

" ... ... and since each \(\displaystyle M_i\) is isomorphic to a submodule of \(\displaystyle \bigoplus_{ i = 1 }^n M_i\) ... ... "Can someone please explain to me how/why each \(\displaystyle M_i\) is isomorphic to a submodule of \(\displaystyle \bigoplus_{ i = 1 }^n M_i\) ... ... ?

Do we know what the submodule is ... ?
Help will be appreciated ...

Peter
What you are asking is the following: If $M$ and $N$ are $R$-modules, then how is $M$ as submodule of $M\oplus N$? Just unravel the definition of $M\oplus N$. $M\oplus N$ as a set is just $M\times N$, where you add and scale coordinate-wise. So a natural embedding of $M$ into $M\oplus N$ is $i:M\to M\oplus N$ given by $i(x)=(x, 0)$. Do you see why $i$ is an injective $R$-module homomorphism?
 
  • #3
caffeinemachine said:
What you are asking is the following: If $M$ and $N$ are $R$-modules, then how is $M$ as submodule of $M\oplus N$? Just unravel the definition of $M\oplus N$. $M\oplus N$ as a set is just $M\times N$, where you add and scale coordinate-wise. So a natural embedding of $M$ into $M\oplus N$ is $i:M\to M\oplus N$ given by $i(x)=(x, 0)$. Do you see why $i$ is an injective $R$-module homomorphism?
Thanks for the help, caffeinemachine ... ...

You write:

" ... ... Do you see why $i$ is an injective $R$-module homomorphism? ... ... "

Well ...

\(\displaystyle i( x + y ) = i(x) + i(y)\) holds because ...

\(\displaystyle i(x+y) = (x+y, 0 ) = (x,0) + (y,0) = i(x) + i(y)\) ... ...

... and ...

\(\displaystyle i(xa) = i(x) a\) holds where \(\displaystyle a \in\) ring \(\displaystyle R\) holds since ...

\(\displaystyle i(xa) = (xa,0) = (x,0) a = i(x) a\) ...

So \(\displaystyle i\) is a homomorphism ...

... and ...

if \(\displaystyle i(x) = i(y)\) ... then ...

\(\displaystyle (x,0) = (y,0) \)

\(\displaystyle \Longrightarrow x = y\)

... so \(\displaystyle i\) is injective ...Peter
 

FAQ: Direct Sums of Noetherian Modules .... Bland Proposition 4.2.7 .... ....

1. What is a direct sum of Noetherian modules?

A direct sum of Noetherian modules is a way of combining two or more Noetherian modules into a single module in a way that preserves their individual properties.

2. What is the significance of Bland Proposition 4.2.7?

Bland Proposition 4.2.7 is a result that shows the relationship between the Noetherian property and direct sums of modules. It states that if two modules are Noetherian, then their direct sum is also Noetherian.

3. How is Bland Proposition 4.2.7 used in mathematics?

Bland Proposition 4.2.7 is used in mathematics to prove the Noetherian property for larger structures by breaking them down into smaller Noetherian modules and using the result to show that the larger structure is also Noetherian.

4. What are some examples of Noetherian modules?

Some examples of Noetherian modules include finite-dimensional vector spaces, finitely generated modules over a commutative ring, and modules over a principal ideal domain.

5. How does the concept of direct sums of Noetherian modules relate to other areas of mathematics?

The concept of direct sums of Noetherian modules has applications in various areas of mathematics, such as algebraic geometry, commutative algebra, and representation theory. It also plays a role in the study of finite-dimensional vector spaces and finite groups.

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