Maximal Ideal in Simple Ring: Understanding the Relationship Between N and R/N

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In summary, a simple ring is a type of ring in abstract algebra with only the zero ideal and the entire ring itself. A maximal ideal is an ideal that is not properly contained in any other ideal. Simple rings have a unique maximal ideal, while every maximal ideal in a ring is a simple ring. Not all maximal ideals are simple, and a ring can have more than one maximal ideal, particularly in a commutative ring where every proper ideal is contained within a maximal ideal.
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tinynerdi
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Homework Statement


how that N is a maximal ideal in a ring R if and only if R/N is a simple ring. that is it is nontrivial and has no proper nontrivial ideals.


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The Attempt at a Solution


I don't know how to start. Please help.
 
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  • #2
Hint: if you have an ideal in R/N, what do you get by taking its inverse image under the quotient map?
 
  • #3
it is R?
 

1. What is a simple ring?

A simple ring is a type of ring in abstract algebra that does not have any non-trivial (non-zero and non-identity) two-sided ideals. This means that the only ideals of a simple ring are the zero ideal and the entire ring itself.

2. What is a maximal ideal?

A maximal ideal is an ideal of a ring that is not properly contained in any other ideal (except for the entire ring). In other words, if you were to add any element outside of the maximal ideal to the ideal, it would no longer be an ideal.

3. How are simple rings and maximal ideals related?

In a simple ring, every non-zero ideal is maximal. This means that simple rings have a unique maximal ideal (the entire ring itself). Conversely, every maximal ideal in a ring is a simple ring.

4. Are all maximal ideals simple?

No, not all maximal ideals are simple. For example, in the ring of integers (Z), the ideal generated by 2 (2Z) is a maximal ideal, but it is not a simple ring. This is because it has non-trivial ideals (such as 4Z) contained within it.

5. Can a ring have more than one maximal ideal?

Yes, a ring can have more than one maximal ideal. In fact, in a commutative ring, every proper ideal is contained within a maximal ideal. This means that there are many maximal ideals in a commutative ring, and they are all distinct from each other.

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