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:

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

-----------------------------------------------------------------------------

Proof. Let N be an R-Module and suppose that, for each \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 f_\alpha from N to 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

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

 

[Math Amateur]

Still reflecting on this issue Greg ...

Hoping for some help,

Peter

 

[disregardthat]

This is the proof of the universal property of the direct product of modules. We say that for any N and family of maps f_{\alpha} : N \to M_{\alpha}, there is a unique map f : N \to \prod_{\Delta} M_{\alpha} such that the above diagram commute.

So basically, given and N and such a family of maps, you can produce the map f.

In the sense of this theorem you are not given an arbitrary N, and then assuming that such a family of maps exist. There's not a condition merely on N, there's a condition on N and a family of maps f_{\alpha}.

The direct product always exists, by the way, as it is defined above.

 

[Math Amateur]

Thanks disregardthat ... appreciate your help ..

So the point is to show that the direct product exists ... that helps considerably ... now understand the point of the Proposition ... thanks ...

But still somewhat perplexed ... so be grateful if you can help on two issues ...

1. How, or in what way does showing the existence of f demonstrate the existence of the direct product ( \prod_\Delta, M_\alpha \pi_\alpha )?

2. why, in order to show the existence of the the direct product ( \prod_\Delta, M_\alpha \pi_\alpha ) do we need to show that f exists for every R-module N and every family \{ f_\alpha \ : \ N \to M_\alpha \}_\Delta

Hope you can help?

Peter

 

[disregardthat]

Thanks disregardthat ... appreciate your help ..

So the point is to show that the direct product exists ... that helps considerably ... now understand the point of the Proposition ... thanks ...

Just to be clear: The direct product trivially exists, because it is defined above. The point of the theorem is to show the universal (or fundamental) property of the direct product, not that it exists.

1. How, or in what way does showing the existence of f demonstrate the existence of the direct product ( \prod_\Delta, M_\alpha \pi_\alpha )?

As I said above, this is not about showing the existence of the direct product.

2. why, in order to show the existence of the the direct product ( \prod_\Delta, M_\alpha \pi_\alpha ) do we need to show that f exists for every R-module N and every family \{ f_\alpha \ : \ N \to M_\alpha \}_\Delta

Take a look at the text of the theorem and read it carefully. We're proving a specific property of the direct product.

 

[Math Amateur]

Thanks ...

Are you able to say something about why Bland is proving this ... that is, what is the significance and consequences of the theorem ... are you able to comment briefly ...

In (partly) answering this question, Bland says the following on page 41 (the page following Proposition 2.1.1 on page 40) as follows:

"Proposition 2.1.1 and the preceding discussion provide the motivation for the formal definition of a direct product of a family of R-modules."

He then formally defines a direct product as follows:

Note that Bland writes:

"Proposition 2.1.1 shows that every family \{ M_\alpha \}_\Delta of R-modules has a direct product"

This is what got me thinking Bland was using Proposition 2.1.1 to establish the existence of a direct product, but I expressed myself badly ... I think Bland is using proposition 2.1.1 in order to precisely define the direct product ...

BUT ... this leaves me somewhat puzzled ... why is he doing this ... what exactly is wrong or weak or inexact in his previous definition where he opened his discussion of direct products?

Bland's original definition is as follows:

Why not stay with the original definition? What is gained by basing a definition on Proposition 2.1.1?

Can someone please help?

Peter

 

[disregardthat]

Proving the universal property for the direct product, shows that the direct product of modules is a product in the category of R-modules.

In any category (not only the category of R-modules), products make sense, but they do not always exist. By showing the universal property of (categorical) products for direct products, we have shown that products exist in the category of R-modules. Therefore, in line with a more general (categorical) view, it makes sense to say that a product of R-modules is something which satisfies a certain universal property. Then, using the theorem, we know that it exists.

There are many things in commutative algebra which will be defined via universal properties. Another notable example is the tensor product of two modules. There will always be an existence and a uniqueness (up to isomorphism) proof following such a definition. The existence proof will consist of constructing an object satisfying the universal property. While it is a valid point to ask why we don't just define it by its specific construction, in practice it is much more convenient to treat these objects as something which satisfies a certain universal property not necessarily bound to any given construction (there can be many). There will be so many cases where we have different representations of the product/coproduct/tensor product/limit/colimit (all these are defined by universal properties) so it is more of a hassle to care about specific constructions than to just treat them as defined abstractly.As a side note, I personally think it's a bit odd to call definition 2.1.2 a definition of direct products. As I'm used to, direct product refer to the specific construction, and product (or categorical product) in the category of R-modules as the object defined via the universal product. I'd rather see him call it a product in 2.1.2. I don't agree with calling them both direct products, as this makes it confusing whenever he refers to the "direct product". I may be wrong, but I think this is non-standard terminology. Someone else might chime in on this issue.

 

[Math Amateur]

Thanks disregardthat ... that is an extremely helpful post! Still reflecting over what you wrote ... but you have convinced me that I need to detour from my study of rings and modules and get a basic understanding of category theory ...

I am still thinking over your interesting point regarding Definition 2.1.2 ...

By the way, when you write:

" I'd rather see him call it a product in 2.1.2. I don't agree with calling them both direct products, ... ... ... "

do you mean that you would rather that Bland call P a product in Definition 2.1.2 ... ... or is the 2.1.2 in your sentence referring to Proposition 2.1.2?

The way I am reading what you said is that you are arguing that P in Definition 2.1.2 should be referred to as a 'product' (not a 'direct product') and that the original definition of the construction of a direct product at the beginning of Bland's section on 'Direct Products and Direct Sums' (where he first defines direct product) be regarded as the 'direct product'.

Have I interpreted you correctly?

Peter

 

[disregardthat]

Yes, you interpret correctly.