# Ring Direct Products .... Bland Problem 3(a), Problem Set 2.1, Page 49 ....

• MHB
• Math Amateur
In summary: I see. So, I am not the only one who has strange names ... :) ...In summary, Bland's book Rings and Their Modules provides a summary of direct products and direct sums. He provides an equation that explains how these two concepts are related.
Math Amateur
Gold Member
MHB
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 someone to check my solution to Problem 3(a) of Problem Set 2.1 ...

Problem 3(a) of Problem Set 2.1 reads as follows:https://www.physicsforums.com/attachments/8062

My attempt at a solution follows:We claim that every right ideal of the ring $$\displaystyle R_1 \times R_2 \times \ ... \ ... \ \times R_n$$ is of the form $$\displaystyle A_1 \times A_2 \times \ ... \ ... \ \times A_n$$ ...Proof:

Suppose $$\displaystyle A$$ is a right ideal of $$\displaystyle R_1 \times R_2 \times \ ... \ ... \ \times R_n$$ ...

Let $$\displaystyle a \in A$$ and put $$\displaystyle A_1 = \pi_1 (A) , A_2 = \pi_2 (A) , \ ... \ ... \ , A_n = \pi_n (A)$$
Now $$\displaystyle a \in A$$ ...$$\displaystyle \Longrightarrow \pi_1(a) = a_1, \pi_2(a) = a_2, \ ... \ ... \ , \pi_n(a) = a_n$$

for some $$\displaystyle a_1 \in A_1, a_2 \in A_2, \ ... \ ... \ , a_n \in A_n$$Hence ...

$$\displaystyle a = ( i_1 \pi_1 + i_2 \pi_2 + \ ... \ ... \ + i_n \pi_n ) (a)$$$$\displaystyle = (a_1, 0, 0, \ ... \ ... \ , 0) + (0, a_2, 0, \ ... \ ... \ , 0) + \ ... \ ... \ + ( 0, 0, \ ... \ ... \ , a_n )$$$$\displaystyle = ( a_1, a_2, \ ... \ ... \ , a_n)$$Hence ... $$\displaystyle A \subseteq A_1 \times A_2 \times \ ... \ ... \ \times A_n$$ ... ... ... ... ... (1)
Conversely ... Let $$\displaystyle a_1 \in A_1, a_2 \in A_2, \ ... \ ... \ , a_n \in A_n$$Note that again ... $$\displaystyle A$$ is a right ideal of $$\displaystyle R_1 \times R_2 \times \ ... \ ... \ \times R_n$$ ...

... and $$\displaystyle a \in A$$ and put $$\displaystyle A_1 = \pi_1 (A) , A_2 = \pi_2 (A) , \ ... \ ... \ , A_n = \pi_n (A)$$Then there are $$\displaystyle b_1, b_2, \ ... \ ... \ , b_n \in A$$ such that ...

$$\displaystyle \pi_1 (b_1) = a_1, \pi_2 (b_2) = a_2, \ ... \ ... \ , \pi_n (b_n) = a_n$$ ... Hence ...

$$\displaystyle b_1 ( 1,0,0, \ ... \ ... \ , 0 ) + b_2 ( 0, 1,0, \ ... \ ... \ , 0 ) + \ ... \ ... \ + b_n ( 0, 0,0, \ ... \ ... \ , 1 )$$$$\displaystyle = ( i_1 \pi_1 + i_2 \pi_2 + \ ... \ ... \ + i_n \pi_n ) ( b_1 ( 1,0,0, \ ... \ ... \ , 0 ) + b_2 ( 0, 1,0, \ ... \ ... \ , 0 ) + \ ... \ ... \ + b_n ( 0, 0,0, \ ... \ ... \ , 1 ) )$$$$\displaystyle = ( a_1, a_2, \ ... \ ... \ , a_n)$$So ...$$\displaystyle A_1 \times A_2 \times \ ... \ ... \ \times A_n \subseteq A$$ ... ... ... ... ... (2)Now ... $$\displaystyle (1), (2) \Longrightarrow A = A_1 \times A_2 \times \ ... \ ... \ \times A_n$$
Can someone please critique my proof ... and either confirm it is correct or point out the errors and shortcomings ... ...

Problem/Issue ... there is part of the above proof I do not fully understand ... I will relate the issue to text solution for $$\displaystyle n = 2$$ ...In Bland's text on the problem we read the following:

"... ... Hence $$\displaystyle a(1,0) + b(0,1) = ( i_1 \pi_1 + i_2 \pi_2 ) ( a(1,0) + b(0,1) ) = (a_1, a_2)$$, so $$\displaystyle A_1 \times A_2 \subseteq A$$. ... ... "I have two questions regarding the above quote:(1) Exactly why/how is the equation $$\displaystyle a(1,0) + b(0,1) = ( i_1 \pi_1 + i_2 \pi_2 ) ( a(1,0) + b(0,1) ) = (a_1, a_2)$$ ... true?

Can someone please explain in detail why/how this is true ...
(2) Exactly why/how does the above equation being true imply that $$\displaystyle A_1 \times A_2 \subseteq A$$ ... ?
Help with the above will be much appreciated ...

Peter

Bland uses that $A$ is a right ideal of $R_1 \times R_2$.
So, if $a=(x,y) \in A$ and $u=(r,s) \in R_1 \times R_2$ then $au=(x,y)(r,s)$ exists and belongs to $A$.
Also $a(1,0)$ and $b(0,1)$, where $a,b \in A$, both exist and belong to $A$.
He then proves that $(a_1,a_2)=a(1,0)+b(0,1) \in A$
Now, try it again.

steenis said:
Bland uses that $A$ is a right ideal of $R_1 \times R_2$.
So, if $a=(x,y) \in A$ and $u=(r,s) \in R_1 \times R_2$ then $au=(x,y)(r,s)$ exists and belongs to $A$.
Also $a(1,0)$ and $b(0,1)$, where $a,b \in A$, both exist and belong to $A$.
He then proves that $(a_1,a_2)=a(1,0)+b(0,1) \in A$
Now, try it again.
Hi Steenis ... sorry to be slow in responding ...

But Hugo (one of my friends large standard poodles - Hugo and Pierre...) chewed up my glasses ...

Will be back in touch soon ...

Peter

No worries, I fill my time with studying.

(Hugo and Pierre ?)

steenis said:
No worries, I fill my time with studying.

(Hugo and Pierre ?)
Hugo and Pierre are the names of my friends large standard poodles ...

Peter

The point is: Hugo is my first name and Pierre = Peter is your first name ....

steenis said:
The point is: Hugo is my first name and Pierre = Peter is your first name ....
Quite a coincidence ... :) ...

Peter

Does this make us antipodal poodles ?

steenis said:
Does this make us antipodal poodles ?
Well ... the two standard poodles are intelligent... but quite eccentric ... :) ...

Peter

## 1. What is the "Bland Problem" in Ring Direct Products?

The Bland Problem refers to a mathematical optimization problem that involves finding the minimum or maximum value of a linear objective function, subject to a set of linear constraints. It is named after mathematician Robert J. Bland, who first described the problem in 1977.

## 2. What is Problem 3(a) in Ring Direct Products?

Problem 3(a) is a specific mathematical problem within the Bland Problem that is found in Section 2.1 on Page 49 of the Ring Direct Products textbook. It is an example of a linear programming problem that involves finding the optimal values of a given set of variables.

## 3. How is Problem 3(a) solved in Ring Direct Products?

Problem 3(a) is typically solved using the simplex method, which is an algorithm that systematically explores the feasible region of a linear programming problem to find the optimal solution. In Ring Direct Products, this can be done using various mathematical techniques and software programs.

## 4. What is the purpose of Problem Set 2.1 in Ring Direct Products?

Problem Set 2.1 is a set of exercises found in the Ring Direct Products textbook that are designed to help students practice and apply the concepts and techniques learned in Section 2.1. These problems are meant to reinforce understanding and improve problem-solving skills.

## 5. Are there real-world applications for the concepts in Ring Direct Products' Bland Problem 3(a)?

Yes, the Bland Problem and Problem 3(a) have many real-world applications in fields such as economics, engineering, and operations research. These concepts are used to model and solve various optimization problems, such as resource allocation, production planning, and transportation logistics.

Replies
5
Views
1K
Replies
2
Views
2K
Replies
8
Views
2K
Replies
1
Views
952
Replies
17
Views
2K
Replies
2
Views
1K
Replies
2
Views
2K
Replies
2
Views
1K
Replies
19
Views
5K
Replies
67
Views
11K