Proving IJ is an Ideal When S is Not: A Case Study

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In summary, the conversation discusses the properties of ideals in a commutative ring. It is shown that while the set S, defined as the product of two ideals I and J, may not always be an ideal, the set IJ is always an ideal and is called the product ideal. The conversation also considers the equivalence between S and IJ, and raises questions about whether S can be an ideal.
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Homework Statement


Show that if I and J are ideals of the (commutative) ring; R then
S={xy|x in I y in J} is not necessarily an ideal but the set of finite sums
IJ={Ʃ(xvyv)|xvin I yvin J}
is (and called the product ideal).

Homework Equations


An ideal satisfies the properties
For all x, x' in I, all r in R
(i) x+x' in I
(ii) rx in I


The Attempt at a Solution


This seems wrong to me since if I is generated by i and J is generated by j
xy+x'y'= ai*bj+a'i*b'j=(ab+a'b')(ij)=(ab+a'b')i*j which is in S since (ab+a'b')i is in I and j is in J...
and obviously by the same rule;
r(xy) is in S.
Therefore, it seems to me that S and IJ are equivalent sets (since the elements of S could be split up into a sum of product elements) and I don't see how S could _not_ be an ideal.

I haven't been able to find much information about product ideals; but this is a problem in my textbook - Algebra by Michael Artin (second edition) so I'm disinclined to think the lecturer phrased the question wrong...
Please help?
 
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Related to Proving IJ is an Ideal When S is Not: A Case Study

1. What is the purpose of proving IJ is an ideal when S is not?

The purpose of proving IJ is an ideal when S is not is to understand the properties of ideal rings and their relationship with non-ideal elements. It serves as a case study to further our knowledge and understanding of algebraic structures.

2. Why is it important to study ideal rings?

Ideal rings play a crucial role in abstract algebra and have numerous applications in various fields of mathematics such as number theory, cryptography, and coding theory. Understanding their properties and behavior is essential for solving complex problems and developing new mathematical theories.

3. What does it mean for a ring to be an ideal?

An ideal is a special subset of a ring that satisfies certain conditions, such as closure under addition and multiplication by elements of the ring. It is a fundamental concept in ring theory and is used to define important algebraic structures such as quotient rings and homomorphisms.

4. How is the proof for proving IJ is an ideal when S is not structured?

The proof for proving IJ is an ideal when S is not is structured by first defining the ideal IJ and then showing that it satisfies the necessary conditions to be considered an ideal. This involves using the properties of rings and ideals to demonstrate closure under addition and multiplication by elements of the ring.

5. What are the key takeaways from this case study?

The key takeaways from this case study are understanding the concept of ideal rings and their properties, the relationship between ideal and non-ideal elements, and the importance of proving theorems in abstract algebra. This case study also highlights the application of theoretical concepts in solving real-world problems.

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