Proving Ideal of R/J is Contained in I of R Containing J

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In summary, the ideal of R/J is the set of elements in the ring R that have a remainder of 0 when divided by the ideal J. This ideal is a subset of the ideal of R, meaning that every element in R/J is also in R. It is also contained in the ideal I of R containing J, and this containment can be proved by showing that every element in R/J satisfies the definition of an element in I of R containing J. This is important because it helps to understand the relationship between different ideals in a ring and can provide insights into the structure of the ring and its applications.
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Homework Statement


Suppose that J is an ideal of R, and consider the ring R/J = {r + J | r 2 R}.
Prove that X is an ideal of R/J is and only if there is an ideal I of R containing J such
that J c I c R.


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The Attempt at a Solution

 
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What was your attempt at a solution?
 

1. What exactly is the ideal of R/J?

The ideal of R/J is the set of all elements in the ring R that are congruent to 0 modulo J, where J is a specific ideal of R. In other words, it is the set of all elements in R that, when divided by J, have a remainder of 0.

2. How is the ideal of R/J related to the ideal of R?

The ideal of R/J is a subset of the ideal of R, meaning that every element in the ideal of R/J is also in the ideal of R. This is because every element in R/J must be congruent to 0 modulo J, and therefore must also be in the ideal of R.

3. What does it mean for the ideal of R/J to be contained in I of R containing J?

This means that every element in the ideal of R/J is also in the ideal I of R containing J. In other words, the ideal of R/J is a subset of I of R containing J, and every element in the ideal of R/J is also in I of R containing J.

4. How can the containment of the ideal of R/J in I of R containing J be proved?

This can be proved by showing that every element in the ideal of R/J is also in I of R containing J. This can be done by taking an arbitrary element in the ideal of R/J and showing that it satisfies the definition of an element in I of R containing J (i.e. it is congruent to 0 modulo J).

5. Why is proving the containment of the ideal of R/J in I of R containing J important?

This is important because it helps to understand the relationship between two different ideals in a ring. It also allows us to make conclusions about the structure of the ring and its ideals, which can be useful in various mathematical and scientific applications.

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