Direct Sums and Factor Modules .... Bland Problem 14, Problem Set 2.1

In summary, Steenis provided a helpful summary of the content of Paul E. Bland's book Rings and Their Modules. He provided an obvious R-map between $M$ and $M/N$, and provided a definition for f. The First Isomorphism Theorem showed that f is a surjective homomorphism with kernel $\text{Ker} f$. Steenis also provided helpful suggestions for proceeding with Problem 14 of Problem Set 2.1.
  • #1
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I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ...

I need help to make a meaningful start on Problem 14 of Problem Set 2.1 ...

Problem 14 of Problem Set 2.1 reads as follows:View attachment 8115I am somewhat overwhelmed by this problem ...

Can someone please hep me to make a meaningful start on the problem ...Hope someone can help ...

Peter
 
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  • #2
Define an R-homomorphism $$f:\bigoplus_\Delta M_\alpha \to \bigoplus_\Delta \frac{M_\alpha}{N_\alpha}$$
 
  • #3
steenis said:
Define an R-homomorphism $$f:\bigoplus_\Delta M_\alpha \to \bigoplus_\Delta \frac{M_\alpha}{N_\alpha}$$
Thanks for the help, Steenis ...

Hmm ... but still not sure how to proceed ...

Can you help a bit further ...

Peter
 
  • #4
It is an obvious R-map.

Another hint: what is the obvious R-map between $M$ and $M/N$ ?
 
  • #5
steenis said:
It is an obvious R-map.

Another hint: what is the obvious R-map between $M$ and $M/N$ ?
Hi Steenis ...

The obvious R-map between $M$ and $M/N$ is the canonical surjection or natural map \(\displaystyle \eta \ : \ M \rightarrow M/N\) ...

... defined by \(\displaystyle \eta (x) = x + N = \overline{ x }\) ...
Now you suggest we consider ...

\(\displaystyle f:\bigoplus_\Delta M_\alpha \to \bigoplus_\Delta \frac{M_\alpha}{N_\alpha}\)... but how do we define this R-homomorphism ...?Well ... we can write \(\displaystyle x \in \bigoplus_\Delta M_\alpha\) as \(\displaystyle x = \sum_\Delta x_\alpha\) ...... but how do we define \(\displaystyle f\) ... is it something like ...\(\displaystyle f(x) = f ( \sum_\Delta x_\alpha ) = \sum_\Delta ( x_\alpha + N_\alpha )\) ...Unsure of this ... can you help further ...

... and anyway ... how do we proceed ...I am having trouble understanding the exact and explicit form of elements in \(\displaystyle ( \bigoplus_\Delta M_\alpha ) / ( \bigoplus_\Delta N_\alpha )\) and \(\displaystyle \bigoplus_\Delta M_\alpha / N_\alpha\) ...Hope you can help further ...

Peter
 
  • #6
1) from the context it does not show that the direct sums are internal, but that they are external (compare the definitions), so the notations should be ...

2) if you have the notations right, your R-map is correct.

3) now determine the kernel of the R-map.
 
  • #7
steenis said:
1) from the context it does not show that the direct sums are internal, but that they are external (compare the definitions), so the notations should be ...

2) if you have the notations right, your R-map is correct.

3) now determine the kernel of the R-map.
Hi Steenis ... thanks ...

Have taken note of the points you have made ...We have to show that \(\displaystyle ( \bigoplus_\Delta M_\alpha ) / ( \bigoplus_\Delta N_\alpha )\) \(\displaystyle \cong\) \(\displaystyle \bigoplus_\Delta M_\alpha / N_\alpha\) ...So we define ...

\(\displaystyle f:\bigoplus_\Delta M_\alpha \to \bigoplus_\Delta M_\alpha / N_\alpha \)

where \(\displaystyle f( (x_\alpha) ) = \bigoplus_\Delta ( x_\alpha + N_\alpha ) \)Now f is a surjective homomorphism with kernel ...

\(\displaystyle \text{Ker } f = \bigoplus_\Delta ( 0 + N_\alpha ) = \bigoplus_\Delta N_\alpha\) ...Therefore ... by the First Isomorphism Theorem we have ...\(\displaystyle ( \bigoplus_\Delta M_\alpha ) / \text{Ker } f \) \(\displaystyle \cong\) \(\displaystyle \bigoplus_\Delta M_\alpha / N_\alpha\) ...That is ...\(\displaystyle ( \bigoplus_\Delta M_\alpha ) / ( \bigoplus_\Delta N_\alpha )\) \(\displaystyle \cong\) \(\displaystyle \bigoplus_\Delta M_\alpha / N_\alpha\) ...
Is that correct... ?

Peter
 
  • #8
Yes all correct, one remark:

$\text{Ker } f = \bigoplus_\Delta ( n_\alpha + N_\alpha ) = \bigoplus_\Delta N_\alpha$, where $n_\alpha \in N_\alpha$
 
  • #9
steenis said:
Yes all correct, one remark:

$\text{Ker } f = \bigoplus_\Delta ( n_\alpha + N_\alpha ) = \bigoplus_\Delta N_\alpha$, where $n_\alpha \in N_\alpha$
Thanks for the correction, Steenis ...

... ... and thanks for all your help with this problem ...

Peter
 

Related to Direct Sums and Factor Modules .... Bland Problem 14, Problem Set 2.1

1. What is a direct sum in linear algebra?

A direct sum in linear algebra refers to the combination of two or more subspaces of a vector space that do not share any common elements, resulting in a new vector space that is the direct sum of the original subspaces. This means that any vector in the direct sum can be uniquely expressed as the sum of vectors from the original subspaces.

2. How do you find the direct sum of two subspaces?

To find the direct sum of two subspaces, you need to ensure that the subspaces do not have any common elements. This can be done by checking if the intersection of the subspaces is only the zero vector. If this is the case, then the direct sum can be formed by taking all possible combinations of vectors from each subspace.

3. What is a factor module in abstract algebra?

In abstract algebra, a factor module (also known as a quotient module) is a module that is formed by taking a module and a submodule, and then dividing the module by the submodule. This results in a new module that contains all the cosets of the submodule, and the operations on this new module are defined by the operations on the original module.

4. How do you determine the factor module of a given module and submodule?

To determine the factor module of a given module and submodule, you need to first check if the submodule is a normal submodule, meaning that it is invariant under the module's operations. If it is, then the factor module can be formed by taking all the cosets of the submodule and defining the operations on the factor module based on the operations of the original module.

5. What is the Bland problem in linear algebra?

The Bland problem, also known as the Bland's Theorem, is a problem related to the direct sum of subspaces in linear algebra. It states that if a vector space is the direct sum of two subspaces, then it is also the direct sum of any other pair of subspaces that contain the original subspaces. This allows for various combinations of subspaces to form the same direct sum, making it a problem in finding the most efficient or simplest way to express a vector space as a direct sum of subspaces.

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