Prime Ideals of Z/3Z X Z/9Z

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In summary, the conversation discusses the concept of prime ideals and finding them for certain multiplication rings. The speaker is struggling to start a problem and is seeking guidance on where to begin. They also mention that R/P being an integral domain, and how Z3 x Z9 has unity, which is important for this concept. The conversation also provides a hint regarding finite integral domains and the fields Z9 and Z3 x Z3.
  • #1
shadowstalker
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Having worked on prime ideals recently and finding them for Z_n I was wondering how you can to find all the prime ideals of a multiplication of Z/3Z X Z/9Z or Z/2Z X Z/4Z for example. I'm mostly having trouble starting this problem and feel that if I could get an idea where to start that I could finish off the proof myself.

Thanks
 
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  • #2
if R is a commutative ring, and P is a prime ideal, then R/P is an integral domain (some authors require R to have unity for both concepts, however Z3 x Z9 has unity, namely (1,1), so does Z2 x Z4).

what's another name for a finite integral domain?

(hint #2: is Z9 a field? what about Z3 x Z3? think in terms of possible ring homomorphisms).
 

1. What is a prime ideal in Z/3Z x Z/9Z?

A prime ideal in Z/3Z x Z/9Z is a subset of the ring Z/3Z x Z/9Z that satisfies certain properties. Specifically, it is an ideal that is also a prime element in the ring, meaning that it is not a unit and whenever two elements multiply to give an element in the ideal, at least one of the original elements must also be in the ideal.

2. How many prime ideals are there in Z/3Z x Z/9Z?

There are a total of 6 prime ideals in Z/3Z x Z/9Z. This can be determined by looking at the factorization of the ring, which consists of 3 prime ideals in Z/3Z and 2 prime ideals in Z/9Z.

3. Can a prime ideal contain more than one element?

Yes, a prime ideal can contain multiple elements. In fact, all prime ideals in Z/3Z x Z/9Z contain more than one element, as they are defined to be proper subsets of the ring.

4. How are prime ideals related to prime numbers?

The concept of prime ideals is related to the concept of prime numbers in number theory. Just as a prime number cannot be broken down into smaller factors, a prime ideal cannot be generated by smaller ideals. Additionally, prime ideals are analogous to prime numbers in the sense that they are the building blocks of the ring and can be used to understand the structure of the ring.

5. What is the significance of studying prime ideals in Z/3Z x Z/9Z?

Studying prime ideals in Z/3Z x Z/9Z is important because it allows for a deeper understanding of the structure and properties of the ring. Prime ideals play a crucial role in factorization, which is a key concept in number theory and has numerous applications in mathematics and other fields. Additionally, the study of prime ideals can lead to insights and connections between different areas of mathematics.

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