# Is $$\bigoplus_\Delta A_\alpha$$ a Right Ideal of $$\prod_\Delta R_\alpha$$?

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In summary: Steenis is staying with a friend who doesn't drink, but had a couple of drinks of whisky and soda last night.
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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 someone to check my solution to Problem 2(b) of Problem Set 2.1 ...

Problem 2(b) of Problem Set 2.1 reads as follows:
View attachment 8060
My attempt at a solution follows:We claim that $$\displaystyle \bigoplus_\Delta A_\alpha$$ is a right ideal of $$\displaystyle \prod_\Delta R_\alpha$$ Proof ... Let $$\displaystyle (x_\alpha ) , (y_\alpha ) \in \bigoplus_\Delta A_\alpha$$ and let $$\displaystyle (r_\alpha ) \in \prod_\Delta R_\alpha$$Then $$\displaystyle (x_\alpha ) + (y_\alpha ) = (x_\alpha + y_\alpha )$$ ...

... further ... if $$\displaystyle (x_\alpha )$$ has $$\displaystyle m$$ non-zero components and $$\displaystyle (y_\alpha$$) has $$\displaystyle n$$ non-zero components then $$\displaystyle (x_\alpha + y_\alpha )$$ has at most $$\displaystyle (m+n)$$ non-zero components ... that is $$\displaystyle (x_\alpha + y_\alpha )$$ has only a finite number of non-zero components ...

... so ... since each $$\displaystyle x_\alpha + y_\alpha \in A_\alpha$$ we have that $$\displaystyle (x_\alpha + y_\alpha ) \in \bigoplus_\Delta A_\alpha$$ ...

Hence ... $$\displaystyle (x_\alpha ) + (y_\alpha ) \in \bigoplus_\Delta A_\alpha$$ ... ... ... ... ... (1)
Also ... we have ...

$$\displaystyle (x_\alpha ) (r_\alpha ) = (x_\alpha r_\alpha)$$

... and assuming $$\displaystyle x_\alpha$$ has $$\displaystyle m$$ non-zero components, then $$\displaystyle (x_\alpha r_\alpha$$) has at most $$\displaystyle m$$ non-zero components ...... and ...$$\displaystyle x_\alpha r_\alpha \in A_\alpha$$ since $$\displaystyle A_\alpha$$ is a right ideal of $$\displaystyle R_\alpha$$so $$\displaystyle (x_\alpha r_\alpha) \in \bigoplus_\Delta A_\alpha$$and it follows that $$\displaystyle (x_\alpha) ( r_\alpha) \in \bigoplus_\Delta A_\alpha$$ ... ... ... ... ... (2)$$\displaystyle (1) (2) \Longrightarrow \bigoplus_\Delta A_\alpha$$ is a right ideal of $$\displaystyle \prod_\Delta R_\alpha$$
Can someone please critique my proof either by confirming it to be correct and/or pointing out errors and shortcomings ...

Peter

This is correct, Peter.
Why aren't you drinking XXXX in Victoria?

steenis said:
This is correct, Peter.
Why aren't you drinking XXXX in Victoria?
THanks Steenis ...

... well, staying with a friend who doesn't drink ...

But had a couple of drinks of whisky and soda last night ... :) ...

Peter

## 1. What is a direct product of rings?

A direct product of rings is a mathematical construction that combines two or more rings to create a new ring with certain properties. It is denoted by the symbol ⨯ and is defined as the set of all ordered pairs of elements from each of the original rings, with operations defined component-wise.

## 2. How is the direct product of rings different from the direct sum of rings?

While the direct product and direct sum of rings may seem similar, they have different underlying properties. The direct product is defined as the set of all ordered pairs of elements from each of the original rings, while the direct sum is defined as the set of all possible finite sums of elements from each of the original rings. Additionally, the direct product has a component-wise operation while the direct sum has a coordinate-wise operation.

## 3. What is an ideal in the context of rings?

An ideal in the context of rings is a subset of a ring that is closed under addition, subtraction, and multiplication by elements in the ring. It can be thought of as a "sub-ring" that does not necessarily contain all the elements of the original ring. Ideals are useful in studying the structure and properties of rings.

## 4. Can the direct product of rings have an ideal?

Yes, the direct product of rings can have an ideal. Since the direct product is a ring itself, it can have all the same properties and structures as any other ring, including ideals. In fact, the direct product of rings is often used to construct new ideals by taking the product of ideals from each of the original rings.

## 5. How is the Bland Problem 2(b) related to direct products of rings and ideals?

The Bland Problem 2(b) is a question about whether the direct product of rings and ideals can have a non-zero identity element. In other words, it asks if the direct product can have an element that behaves like a "one" in multiplication. This problem highlights the importance of understanding the structure and properties of direct products of rings and ideals in mathematics.

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