# Simple Modules and Right Annihilator Ideals ....

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In summary: Assume ##a \in \text{ann}_r( R / \mathfrak{m} )##. That means that, for any element of ##R/\mathfrak m##, say ##s+\mathfrak m##, we have$$(s+\mathfrak m)a=0_{R/\mathfrak m}\ \ \ \ (1)$$Hence, in particular, we have$$(1+\mathfrak m)a=0_{R/\mathfrak m}\ \ \ (2)$$(I assume, since the proof refers to ##1##, that ##R## is assumed to contain a multipl
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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}##. "Can someone please explain exactly why ##a \in \text{ann}_r( R / \mathfrak{m} )## implies that ##a + \mathfrak{m} = ( 1 + \mathfrak{m} ) a = 0## ... ... ?

Can someone also explain how this then implies that ##a \in \mathfrak{m}## ... ?Hope someone can help ...

Peter

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Assume ##a \in \text{ann}_r( R / \mathfrak{m} )##. That means that, for any element of ##R/\mathfrak m##, say ##s+\mathfrak m##, we have
$$(s+\mathfrak m)a=0_{R/\mathfrak m}\ \ \ \ \ (1)$$
Hence, in particular, we have
$$(1+\mathfrak m)a=0_{R/\mathfrak m}\ \ \ \ (2)$$
(I assume, since the proof refers to ##1##, that ##R## is assumed to contain a multiplicative identity, ie it is a 'unitary ring')

Now by definition of multiplication in a quotient:
$$(1+\mathfrak m)a=1\times a+\mathfrak m=a+\mathfrak m\ \ \ \ (3)$$
and by (2) this is equal to ##0_{R/\mathfrak m}##, which is ##\mathfrak m##.

Thus
$$a+\mathfrak m=\mathfrak m\ \ \ \ (4)$$
which means that ##a\in\mathfrak m##.

Math Amateur
Thanks Andrew ... most clear and very helpful ...

Peter

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}##. "Can someone please explain exactly why ##a \in \text{ann}_r( R / \mathfrak{m} )## implies that ##a + \mathfrak{m} = ( 1 + \mathfrak{m} ) a = 0## ... ... ?

Can someone also explain how this then implies that ##a \in \mathfrak{m}## ... ?Hope someone can help ...

Peter

## 1. What is a simple module?

A simple module is a module that has no proper nonzero submodules. This means that the only submodules of a simple module are itself and the zero module.

## 2. How do you determine the right annihilator ideal of a module?

The right annihilator ideal of a module is the set of all elements in the module's ring that annihilate the module, meaning that when multiplied by any element in the module, the result is always zero. This can be determined by finding the set of all elements in the ring that commute with all elements in the module.

## 3. What is the relationship between simple modules and right annihilator ideals?

In general, a simple module will have a nontrivial right annihilator ideal. However, the converse is not always true - a module with a nontrivial right annihilator ideal may not necessarily be simple.

## 4. Can a simple module have more than one right annihilator ideal?

No, a simple module can only have one right annihilator ideal. This is because a simple module has no proper nonzero submodules, so any element that annihilates the module will also necessarily annihilate any potential right annihilator ideal.

## 5. How do simple modules and right annihilator ideals relate to the structure of a ring?

Simple modules and right annihilator ideals are important concepts in the study of ring theory, as they provide insight into the structure of a ring. In particular, the right annihilator ideal of a simple module can help determine the structure of the ring, as it is a maximal right ideal and can be used to construct other important ideals such as the Jacobson radical.

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