# Jacobson Radical and Right Annihilator Ideals ....

• I
• Math Amateur
In summary, the conversation discusses reading Paul E. Bland's book "Rings and Their Modules" and focusing on Section 6.1 about the Jacobson Radical. The individual needs help with understanding the proof for Proposition 6.1.7, specifically why it follows that the intersection of the set of annihilators of all simple modules is a subset of J(R). The individual also shares their own thoughts on the matter.
Math Amateur
Gold Member
MHB
I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 6.1 The Jacobson Radical ... ...

I need help with the proof of Proposition 6.1.7 ...Proposition 6.1.7 and its proof read as follows:

In the above text from Bland ... in the proof of (1) we read the following:" ... ... we see that ##\text{ann}_r( R / \mathfrak{m} ) = \text{ann}_r(S)##. But ##a \in \text{ann}_r( R / \mathfrak{m} )## implies that ##a + \mathfrak{m} = ( 1 + \mathfrak{m} ) a = 0## , so ##a \in \mathfrak{m}##.

So, it follows that ##\bigcap_\mathcal{S} \text{ann}_r(S) \subseteq J(R)## ... ... "
Could someone please explain why it follows that ##\bigcap_\mathcal{S} \text{ann}_r(S) \subseteq J(R)## ... ... ?Hope someone can help ...

Peter

#### Attachments

• Bland - 1 - Proposition 6.1.7 ... ... PART 1.png
79.8 KB · Views: 710
• Bland - 2 - Proposition 6.1.7 ... ... PART 2.png
42.9 KB · Views: 736
Math Amateur said:
I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 6.1 The Jacobson Radical ... ...

I need help with the proof of Proposition 6.1.7 ...Proposition 6.1.7 and its proof read as follows:

In the above text from Bland ... in the proof of (1) we read the following:" ... ... we see that ##\text{ann}_r( R / \mathfrak{m} ) = \text{ann}_r(S)##. But ##a \in \text{ann}_r( R / \mathfrak{m} )## implies that ##a + \mathfrak{m} = ( 1 + \mathfrak{m} ) a = 0## , so ##a \in \mathfrak{m}##.

So, it follows that ##\bigcap_\mathcal{S} \text{ann}_r(S) \subseteq J(R)## ... ... "
Could someone please explain why it follows that ##\bigcap_\mathcal{S} \text{ann}_r(S) \subseteq J(R)## ... ... ?Hope someone can help ...

Peter
Just some thoughts on my own question ... ...Just some thoughts ...

Since ##\text{ann}_r( R / \mathfrak{m} ) = \text{ann}_r(S)##

we have ##a \in \text{ann}_r( R / \mathfrak{m} )## means ##a \in \text{ann}_r(S)## ...

thus ##a \in \bigcap_\mathscr{S} \text{ann}_r(S)## ... ...

But ... we also have that ##a \in \text{ann}_r( R / \mathfrak{m} )## implies that ##a \in \mathfrak{m}## ... ...

But this means that ##a \in J(R)## ...

Thus ##a \in \bigcap_\mathscr{S} \text{ann}_r(S) \Longrightarrow a \in J(R)## ... ...

So ##\bigcap_\mathscr{S} \text{ann}_r(S) \subseteq J(R)## ...Is that correct?Peter

## What is the Jacobson Radical of an ideal?

The Jacobson Radical of an ideal is defined as the intersection of all maximal left ideals containing the ideal. In other words, it is the largest left ideal contained in the ideal.

## What is the significance of the Jacobson Radical in ring theory?

The Jacobson Radical is significant in ring theory because it measures the "non-commutative" part of a ring. It helps to characterize the structure and properties of a ring, and is used in various theorems and proofs in ring theory.

## How is the Jacobson Radical related to the right annihilator ideal?

The right annihilator ideal of a subset S of a ring R is defined as the set of all elements in R that annihilate S, meaning that multiplying any element of S by an element of the right annihilator will result in 0. The Jacobson Radical is a subset of the right annihilator ideal, and in some cases, they can be equal.

## Can a ring have more than one Jacobson Radical?

No, a ring can have only one Jacobson Radical. This is because the Jacobson Radical is defined as the intersection of all maximal left ideals, and a ring can have only one maximal left ideal.

## What are some applications of Jacobson Radical and Right Annihilator Ideals in mathematics?

The Jacobson Radical and Right Annihilator Ideals have applications in various areas of mathematics, such as representation theory, algebraic geometry, and functional analysis. They are also used in the study of non-commutative rings and modules, and in the classification of rings and algebras.

Replies
1
Views
1K
Replies
3
Views
1K
Replies
1
Views
1K
Replies
6
Views
2K
Replies
8
Views
2K
Replies
10
Views
2K
Replies
7
Views
2K
Replies
3
Views
2K
Replies
4
Views
2K
Replies
25
Views
3K