Math Amateur
Gold Member
MHB
- 3,920
- 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 need help with Problem 1(b) of Problem Set 2.1 ...
Problem 1(b) of Problem Set 2.1 reads as follows:
View attachment 8048I have had difficulty in formulating a rigorous and convincing proof of the statement in Problem 1(b) ... can someone please
(1) critique my attempt at a proof (see below)
(2) provide an alternate rigorous and convincing proof My attempt at a proof is as follows:
We need to demonstrate that $$\prod_\Delta M_\alpha = \bigoplus_\Delta M_\alpha$$ if and only if $$\Delta$$ is a finite set ...Assume $$\prod_\Delta M_\alpha = \bigoplus_\Delta M_\alpha $$
The above equality would require all of the terms $$(x_\alpha)$$ of $$\prod_\Delta M_\alpha$$ to have a finite number of components or elements in each $$(x_\alpha)$$ ... thus $$\Delta$$ is a finite set ...
Assume $$\Delta$$ is a finite set
... then $$\prod_\Delta M_\alpha$$ has terms of the form $$(x_\alpha) = ( x_1, x_2, \ ... \ ... \ , x_n )$$ for some $$n \in \mathbb{Z}$$ ... ...
and
... $$\bigoplus_\Delta M_\alpha$$ has the same terms given that each of the above terms $$(x_\alpha)$$ has a finite number of components ...
Hope someone can indicate how to formulate a better proof ...
Peter
I need help with Problem 1(b) of Problem Set 2.1 ...
Problem 1(b) of Problem Set 2.1 reads as follows:
View attachment 8048I have had difficulty in formulating a rigorous and convincing proof of the statement in Problem 1(b) ... can someone please
(1) critique my attempt at a proof (see below)
(2) provide an alternate rigorous and convincing proof My attempt at a proof is as follows:
We need to demonstrate that $$\prod_\Delta M_\alpha = \bigoplus_\Delta M_\alpha$$ if and only if $$\Delta$$ is a finite set ...Assume $$\prod_\Delta M_\alpha = \bigoplus_\Delta M_\alpha $$
The above equality would require all of the terms $$(x_\alpha)$$ of $$\prod_\Delta M_\alpha$$ to have a finite number of components or elements in each $$(x_\alpha)$$ ... thus $$\Delta$$ is a finite set ...
Assume $$\Delta$$ is a finite set
... then $$\prod_\Delta M_\alpha$$ has terms of the form $$(x_\alpha) = ( x_1, x_2, \ ... \ ... \ , x_n )$$ for some $$n \in \mathbb{Z}$$ ... ...
and
... $$\bigoplus_\Delta M_\alpha$$ has the same terms given that each of the above terms $$(x_\alpha)$$ has a finite number of components ...
Hope someone can indicate how to formulate a better proof ...
Peter