Direct Products of Modules - Bland - Proposition 2.1.1 and its proof

[Math Amateur]

I am reading Paul E. Bland's book, Rings and Their Modules, Section 2.1: Direct Products and Direct Sums.

I have a question regarding the proof of Proposition 2.1.1

Proposition 2.1.1 and its proof (together with with a relevant preliminary definition) read as follows:

https://www.physicsforums.com/attachments/2427

As can be seen in the above text, the first line of the proof reads as follows:

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Proof. Let N be an R-Module and suppose that, for each \(\displaystyle \alpha \in \Delta, \ \ f_\alpha : \ N \to M_\alpha \) is an R-linear mapping.

... ... ... etc. etc.-----------------------------------------------------------------------------

My question is as follows:

How do we know such a family of module homomorphisms or R-linear mappings \(\displaystyle f_\alpha \) from \(\displaystyle N \) to \(\displaystyle M_\alpha \) exist?

... ... or is the point that if they do not exist, then the direct product does not exist?

Hope someone can help.

Peter

 

[Deveno]

Yes, you cannot take the direct product of non-existent mappings.

 

[Math Amateur]

Yes, you cannot take the direct product of non-existent mappings.

Thanks Deveno ... gives me confidence to have your confirmation ...

However, I am still perplexed as to why Bland seeks to establish a unique mapping from every R-module to establish the existence of a direct product ... indeed he also involves every set of R-linear mappings??

Can you help?

Peter

 

[Math Amateur]

Thanks Deveno ... gives me confidence to have your confirmation ...

However, I am still perplexed as to why Bland seeks to establish a unique mapping from every R-module to establish the existence of a direct product ... indeed he also involves every set of R-linear mappings??

Can you help?

Peter

I have now learned that Proposition 2.1.1 actually proves the Universal Mapping Property for the direct product.

However I am still puzzled about Bland's motive for doing this ... but he clearly uses this proof to formally define direct products ... but why does he do this ... indeed ... what was wrong with his 'informal definition' when he opened his discussion of direct products, as follows:https://www.physicsforums.com/attachments/2446The above seems a perfectly good definition to me ... so why does Bland go through Proposition 2.1.1, apparently in order to give a 'formal' or better definition ... in what way is the new definition more exact or better ...

To give MHB members necessary relevant information to see why a new definition may be necessary, here is the 'formal' definition or re-definition of direct product based on consideration of Proposition 2.1.1:View attachment 2447
My question, as indicated above is as follows:

Why is Bland doing this re-definition ... what exactly is wrong or weak or inexact in his previous definition where he opened his discussion of direct products?

I would really appreciate help?

Peter

 

[Deveno]

There's nothing wrong with it, per se. It's just that the universal property is more "portable" in talking about OTHER structures, such as groups, rings, vector spaces (and perhaps more importantly) topological spaces.

The construction is, then, CATEGORICAL, and if such a diagram exists in a given category, one says the category "has products". This is often very useful, and one knows all the important properties of the product without having to re-prove them in the different settings.

More of math (for good or ill) is moving in this direction.

 

[Math Amateur]

There's nothing wrong with it, per se. It's just that the universal property is more "portable" in talking about OTHER structures, such as groups, rings, vector spaces (and perhaps more importantly) topological spaces.

The construction is, then, CATEGORICAL, and if such a diagram exists in a given category, one says the category "has products". This is often very useful, and one knows all the important properties of the product without having to re-prove them in the different settings.

More of math (for good or ill) is moving in this direction.

Thanks Deveno ... seems the implication of what you are saying is that I should make a serious detour in my study of rings and modules and learn some category theory ...

Thanks again,

Peter

 

[Deveno]

Not necessarily...but know when something is characterized by a "universal (mapping) property" that somewhere category theory is lurking beneath it.

When one studies rings, a great deal of focus is on properties of ideals. Most (if not all) of this, can be phrased in terms of surjective ring homomorphisms and kernels, eliminating the need to talk about cosets at all! Is this desirable? It depends on your point of view.

Do you want to see "the big picture" or get well-acquainted with the peculiarities of individual structures? I cannot (and should not) make that choice for you.