- #1

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## Homework Statement

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:

Bland Problem 1, Section 2.1 reads as follows:

I need some help with 1(b) ...

## Homework Equations

The definition of External Direct Sum is as follows:

## The Attempt at a Solution

I 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