How Can You Find Prime Ideals in Products Like Z/3Z x Z/9Z?

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SUMMARY

This discussion focuses on finding prime ideals in products of rings, specifically Z/3Z x Z/9Z and Z/2Z x Z/4Z. It establishes that for a commutative ring R with a prime ideal P, the quotient R/P is an integral domain, which is applicable to both Z/3Z x Z/9Z and Z/2Z x Z/4Z since they possess unity. The conversation also hints at exploring the properties of finite integral domains and the nature of Z/9 as a field, prompting further investigation into ring homomorphisms.

PREREQUISITES
  • Understanding of commutative rings and prime ideals
  • Knowledge of integral domains and their properties
  • Familiarity with finite fields and ring homomorphisms
  • Basic concepts of modular arithmetic and its applications
NEXT STEPS
  • Research the structure of finite integral domains and their characteristics
  • Study the properties of Z/9Z and determine if it is a field
  • Explore ring homomorphisms and their implications in prime ideal theory
  • Investigate the prime ideals of Z/2Z x Z/4Z and their relationships
USEFUL FOR

Mathematicians, algebraists, and students studying ring theory, particularly those interested in prime ideals and integral domains in modular arithmetic.

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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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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).
 

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