# Checking My Solution to Problem 2(c) of Problem Set 2.1: Seeking Critique

• MHB
• Math Amateur
In summary, the conversation was about checking a solution to a problem involving direct products and direct sums of rings. The problem was whether $\bigoplus_\Delta R_\alpha$ is a right ideal of $\prod_\Delta R_\alpha$. The solution provided involved proving that $\bigoplus_\Delta R_\alpha$ is a two-sided ideal of $\prod_\Delta R_\alpha$ by showing that it satisfies the definition of a right ideal and using the fact that each $R_\alpha$ is a right ideal of itself.
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 2(c) of Problem Set 2.1 ...

Problem 2(c) of Problem Set 2.1 reads as follows:View attachment 8061My attempt at a solution follows:We claim that $$\displaystyle \bigoplus_\Delta R_\alpha$$ is a right ideal of $$\displaystyle \prod_\Delta R_\alpha$$ Proof ... Let $$\displaystyle (x_\alpha ) , (y_\alpha ) \in \bigoplus_\Delta R_\alpha$$ and let $$\displaystyle (r_\alpha ) \in \prod_\Delta R_\alpha$$Then $$\displaystyle (x_\alpha ) + (y_\alpha ) = (x_\alpha + y_\alpha )$$ ... by the rule of addition in direct products ...Now ... $$\displaystyle x_\alpha + y_\alpha \in R_\alpha$$ for all $$\displaystyle \alpha \in \Delta$$ ... by closure of addition in rings ... Thus $$\displaystyle (x_\alpha + y_\alpha ) \in \prod_\Delta R_\alpha$$ ...... but also ... since $$\displaystyle (x_\alpha$$) and $$\displaystyle (y_\alpha )$$ each have only a finite number of non-zero components ...

... we have that $$\displaystyle (x_\alpha + y_\alpha )$$ has only a finite number of non-zero components ...

... so ... $$\displaystyle (x_\alpha + y_\alpha ) \in \bigoplus_\Delta R_\alpha$$ ...

Hence $$\displaystyle (x_\alpha ) + (y_\alpha ) \in \bigoplus_\Delta R_\alpha$$ ... ... ... ... ... (1)
Now we also have that ... $$\displaystyle (x_\alpha ) (r_\alpha ) = (x_\alpha r_\alpha)$$ ... ... rule of multiplication in a direct product ...

Now ... $$\displaystyle x_\alpha r_\alpha \in R_\alpha$$ for all $$\displaystyle \alpha \in \Delta$$ ... since a ring is closed under multiplication ...

and ...

$$\displaystyle (x_\alpha r_\alpha)$$ has only a finite number of non-zero components since $$\displaystyle (x_\alpha )$$ has only a finite number of non-zero components ...

So ... $$\displaystyle (x_\alpha r_\alpha) \in \bigoplus_\Delta R_\alpha$$

$$\displaystyle \Longrightarrow (x_\alpha) (r_\alpha) \in \bigoplus_\Delta R_\alpha$$ ... ... ... ... ... (2)
$$\displaystyle (1) (2) \Longrightarrow$$ $$\displaystyle \bigoplus_\Delta R_\alpha$$ is a right ideal of $$\displaystyle \prod_\Delta R_\alpha$$
Can someone please critique my proof ... ... and either confirm its correctness or point out the errors and shortcomings ...

Such help will be much appreciated ...

Peter

Last edited:
This is correct, Peter.
You can also say: $R_\alpha$ is a right ideal of $R_\alpha$, then apply (b).
But you forgot something.

(c) asked is $\bigoplus_\Delta R_\alpha$ an ideal of $\prod_\Delta R_\alpha$ ?

So you have to prove that $\bigoplus_\Delta R_\alpha$ a two-sided ideal of $\prod_\Delta R_\alpha$

## 1. What is the purpose of checking my solution to Problem 2(c) of Problem Set 2.1?

The purpose of checking your solution is to ensure that your answer is correct and to identify any errors or mistakes that may have been made. This is an important step in the problem-solving process as it helps to improve your understanding and mastery of the subject.

## 2. How can I check my solution to Problem 2(c)?

You can check your solution by comparing it to the given answer, asking for feedback from peers or instructors, or using a different method or approach to solve the problem and see if you get the same result.

## 3. What should I do if my solution is incorrect?

If your solution is incorrect, you should review your work and try to identify where you made a mistake. You can also seek help from peers or instructors to understand the correct solution and learn from your mistakes.

## 4. Is it important to seek critique for my solution?

Yes, seeking critique is an essential part of the problem-solving process. It allows you to get feedback and improve your understanding of the subject. It also helps you to identify any errors or gaps in your knowledge and improve your problem-solving skills.

## 5. How can I use the critique to improve my problem-solving skills?

You can use the critique to identify your strengths and weaknesses and work on improving them. You can also ask for specific feedback on areas you struggle with and practice more to improve. Additionally, you can learn from the methods and approaches used by others to solve the problem and apply them in future problem-solving tasks.

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