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I am reading Paul E. Bland's book, Rings and Their Modules.
In Section 2.1: Direct Products and Direct Sums, Bland defines the direct product of a family of modules. He then, in Proposition 2.1.1 shows that there is a unique module homomorphism (or R-Linear mapping) from any particular R-module N to the direct product. He then re-defines the direct product.
My question is - what is going on? Why is Bland doing this?
Can anyone help explain what is going on here? What is the point of demonstrating that there is a unique homomorphism from any module to a given direct product? Further, why would one need to or want to demonstrate that every R-Module maps uniquely onto a direct product? Is this some way to show that the formally defined direct product actually exists? But even then how does the unique homomorphism assure this? What is the motivation for the proposition and the re-definition of the direct product?
Details of Bland's definition, proposition and re-definition follow.
The (first) definition of a direct product of modules is as follows:
Proposition 2.1.1, preceded by an important definition, is as follows:
Finally, the following is Bland's re-definition of direct product in the light of Proposition 2.1.1.
Again, my question is - what is going on here? What is the motivation for Proposition 2.1.1 and what is achieved by the Proposition and the subsequent redefinition?
Hope someone can throw some light on what Bland is doing.
Peter
In Section 2.1: Direct Products and Direct Sums, Bland defines the direct product of a family of modules. He then, in Proposition 2.1.1 shows that there is a unique module homomorphism (or R-Linear mapping) from any particular R-module N to the direct product. He then re-defines the direct product.
My question is - what is going on? Why is Bland doing this?
Can anyone help explain what is going on here? What is the point of demonstrating that there is a unique homomorphism from any module to a given direct product? Further, why would one need to or want to demonstrate that every R-Module maps uniquely onto a direct product? Is this some way to show that the formally defined direct product actually exists? But even then how does the unique homomorphism assure this? What is the motivation for the proposition and the re-definition of the direct product?
Details of Bland's definition, proposition and re-definition follow.
The (first) definition of a direct product of modules is as follows:
Proposition 2.1.1, preceded by an important definition, is as follows:
Finally, the following is Bland's re-definition of direct product in the light of Proposition 2.1.1.
Again, my question is - what is going on here? What is the motivation for Proposition 2.1.1 and what is achieved by the Proposition and the subsequent redefinition?
Hope someone can throw some light on what Bland is doing.
Peter
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