# Help with Problem 2(a), Rings & Modules, Paul E. Bland

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In summary, the solution to problem 2(a) is to show that \prod_\Delta A_\alpha is a right ideal of \prod_\Delta R_\alpha.
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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 2(a) of Problem Set 2.1 ...

Problem 2(a) of Problem Set 2.1 reads as follows:
View attachment 8049I am unsure of my solution to problem 2(a) and need help in the following way ...

... could someone please confirm my solution is correct and/or point out errors and shortcomings ...

... indeed I would be grateful if someone could critique my solution ...
My attempted solution to problem 2(a) is as follows:... we have to show that $$\displaystyle \prod_\Delta A_\alpha$$ is a right ideal of $$\displaystyle \prod_\Delta R_\alpha$$ ...

To demonstrate this we have to show that $$\displaystyle \prod_\Delta A_\alpha$$ is closed under addition and closed under multiplication on the right by an element of $$\displaystyle \prod_\Delta R_\alpha$$ ...So ... let $$\displaystyle (x_\alpha), (y_\alpha) \in \prod_\Delta A_\alpha$$ and $$\displaystyle (r_\alpha) \in \prod_\Delta R_\alpha$$

Then $$\displaystyle x_\alpha, y_\alpha \in A_\alpha$$ for all $$\displaystyle \alpha \in \Delta$$

$$\displaystyle \Longrightarrow x_\alpha + y_\alpha \in A_\alpha$$ since $$\displaystyle A_\alpha$$ is a right ideal of $$\displaystyle R_\alpha$$ for all $$\displaystyle \alpha \in \Delta$$ ...

$$\displaystyle \Longrightarrow (x_\alpha) + (y_\alpha) \in \prod_\Delta A_\alpha$$

$$\displaystyle \Longrightarrow \prod_\Delta A_\alpha$$ is closed under addition ...
Now ... $$\displaystyle (x_\alpha) \in \prod_\Delta A_\alpha , (r_\alpha) \in \prod_\Delta R_\alpha$$

$$\displaystyle \Longrightarrow x_\alpha \in A_\alpha , r_\alpha \in R_\alpha$$ for all $$\displaystyle \alpha \in \Delta$$ ...

$$\displaystyle \Longrightarrow x_\alpha r_\alpha \in A_\alpha$$ since $$\displaystyle A_\alpha$$ is a right ideal of $$\displaystyle R_\alpha$$ ...

$$\displaystyle \Longrightarrow ( x_\alpha r_\alpha ) \in \prod_\Delta A_\alpha$$Thus $$\displaystyle \prod_\Delta A_\alpha$$ is a right ideal of $$\displaystyle \prod_\Delta R_\alpha$$ ...

Hope the above is correct ...

Peter

Correct, but one remark:

Because $x_\alpha + y_\alpha \in A_\alpha$

we have $(x_\alpha + y_\alpha) \in \prod_\Delta A_\alpha$

and $(x_\alpha + y_\alpha) = (x_\alpha) + (y_\alpha)$

and therefore, and so on

Last edited:
steenis said:
Correct, but one remark:

Because $x_\alpha + y_\alpha \in A_\alpha$

we have $(x_\alpha + y_\alpha) \in \prod_\Delta A_\alpha$

and $(x_\alpha + y_\alpha) = (x_\alpha) + (y_\alpha)$

and therefore, and so on
Thanks steenis ... appreciate the help and guidance ...

Peter

## 1. What is the concept of rings and modules?

Rings and modules are mathematical structures that are used to study the properties of algebraic systems. A ring is a set of elements with two operations, addition and multiplication, that follow certain rules. A module is a generalization of a vector space, where the scalars come from a ring instead of a field.

## 2. What is the significance of studying rings and modules?

Rings and modules have important applications in many areas of mathematics and science, such as abstract algebra, number theory, algebraic geometry, and physics. They also provide a framework for understanding more complex algebraic structures.

## 3. What is the difference between a ring and a field?

A ring has two operations, addition and multiplication, while a field has these two operations as well as division. In a field, every non-zero element has a multiplicative inverse, while in a ring, this is not always the case.

## 4. What are some examples of rings and modules?

Examples of rings include the integers, real numbers, and polynomials. Examples of modules include vector spaces, ideals in rings, and submodules of modules.

## 5. How can rings and modules be applied in real-life situations?

Rings and modules have various applications in real-life situations, such as coding theory, cryptography, and signal processing. They are also used in areas such as economics, chemistry, and computer science.

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