- #1

Math Amateur

Gold Member

MHB

- 3,998

- 48

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 am trying to fully understand Bland's definition of a direct product ... and to understand the motivation for the definition ... and the implications of the definition ... BUT ... ... I am finding it a real struggle ... ... ... ...

Bland uses Proposition 2.1.1 to define the direct product \(\displaystyle \prod_\Delta M_\alpha\) ...

I will provide the text of Proposition 2.1.1 in this post ... and to give MHB readers the full context and notation, I will also provide the following (at the end of the post):

* Bland's brief introductory section on direct products

* Bland's actual definition based on Proposition 2.1.1

* a Previous section from Chapter 0 where Bland explains the basics of his notation

The text of Proposition 2.1.1 reads as follows:

View attachment 4892

I find Proposition 2.1.1 (on which the definition of a direct product is based - see actual definition below) a rather puzzling and obscurely abstract way of defining a direct product ... I even find it difficult to frame questions about it ... BUT ... here are some questions coming out of a state of perplexity ... ... ... ...

(i) what is the point of defining the direct product \(\displaystyle \prod_\Delta M_\alpha\) this way ... ... ?(ii) why is it important that a map \(\displaystyle f\) exist from

\(\displaystyle ( x_\alpha ) + ( y_\alpha ) = ( x_\alpha + y_\alpha )\)

and

\(\displaystyle ( x_\alpha ) a = ( x_\alpha a )\)

With reference to the above ... my question is as follows:

Do we regard the symbol \(\displaystyle \prod_\Delta M_\alpha\) as an object with no preliminary structure (which we could have simply called \(\displaystyle P\)) ... ... ... OR ... ... ... do we regard \(\displaystyle \prod_\Delta M_\alpha \)as a Cartesian product of a family of modules made into an \(\displaystyle R\)-module by componentwise addition and scalar multiplication ... ... ... in other words ... ... does \(\displaystyle \prod_\Delta M_\alpha\) go into the proposition with the structure I have just described ... or does the structure come out of Proposition 2.1.1 ...===========================================================

===========================================================

Bland's actual definition of a direct product of a family of modules (based on Proposition 2.1.1) reads as follows:

View attachment 4893

Bland's introduction to direct products reads as follows:View attachment 4894

Bland's explanation of his notation in Chapter 0 reads as follows:http://mathhelpboards.com/attachments/linear-abstract-algebra-14/4895-paul-e-blands-definition-direct-product-modules-category-oriented-definition-bland-basics-notation-ch-0-png

Any help with the above issues will be very much appreciated ... ...

Peter

I am trying to fully understand Bland's definition of a direct product ... and to understand the motivation for the definition ... and the implications of the definition ... BUT ... ... I am finding it a real struggle ... ... ... ...

Bland uses Proposition 2.1.1 to define the direct product \(\displaystyle \prod_\Delta M_\alpha\) ...

I will provide the text of Proposition 2.1.1 in this post ... and to give MHB readers the full context and notation, I will also provide the following (at the end of the post):

* Bland's brief introductory section on direct products

* Bland's actual definition based on Proposition 2.1.1

* a Previous section from Chapter 0 where Bland explains the basics of his notation

The text of Proposition 2.1.1 reads as follows:

View attachment 4892

I find Proposition 2.1.1 (on which the definition of a direct product is based - see actual definition below) a rather puzzling and obscurely abstract way of defining a direct product ... I even find it difficult to frame questions about it ... BUT ... here are some questions coming out of a state of perplexity ... ... ... ...

**Questions**(i) what is the point of defining the direct product \(\displaystyle \prod_\Delta M_\alpha\) this way ... ... ?(ii) why is it important that a map \(\displaystyle f\) exist from

__\(\displaystyle R\)-Module \(\displaystyle N\) ... ... ?(iii) is the ... "__**every**__R-module N" ... part of the definition the reason that this property is sometimes referred to as the__**every**__Mapping Property? (iv) Proposition 2.1.2 uses the symbol \(\displaystyle \prod_\Delta M_\alpha\) suggesting a family of modules \(\displaystyle \{ M_\alpha \}\) together with a Cartesian product plus componentwise addition and scalar multiplication defined as__**Universal**\(\displaystyle ( x_\alpha ) + ( y_\alpha ) = ( x_\alpha + y_\alpha )\)

and

\(\displaystyle ( x_\alpha ) a = ( x_\alpha a )\)

With reference to the above ... my question is as follows:

Do we regard the symbol \(\displaystyle \prod_\Delta M_\alpha\) as an object with no preliminary structure (which we could have simply called \(\displaystyle P\)) ... ... ... OR ... ... ... do we regard \(\displaystyle \prod_\Delta M_\alpha \)as a Cartesian product of a family of modules made into an \(\displaystyle R\)-module by componentwise addition and scalar multiplication ... ... ... in other words ... ... does \(\displaystyle \prod_\Delta M_\alpha\) go into the proposition with the structure I have just described ... or does the structure come out of Proposition 2.1.1 ...===========================================================

===========================================================

Bland's actual definition of a direct product of a family of modules (based on Proposition 2.1.1) reads as follows:

View attachment 4893

Bland's introduction to direct products reads as follows:View attachment 4894

Bland's explanation of his notation in Chapter 0 reads as follows:http://mathhelpboards.com/attachments/linear-abstract-algebra-14/4895-paul-e-blands-definition-direct-product-modules-category-oriented-definition-bland-basics-notation-ch-0-png

Any help with the above issues will be very much appreciated ... ...

Peter

Last edited: