MHB Prime Elements in Non-Integral Domains?

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SUMMARY

The discussion centers on the definition of prime elements in non-integral domains, referencing Dummit and Foote's "Abstract Algebra." It establishes that while the definition of a prime element applies to commutative rings, the presence of zero-divisors complicates their utility. Specifically, the theorem states that for a commutative ring \( R \) and an ideal \( J \), \( J \) is prime if and only if \( R/J \) is an integral domain. Thus, prime elements can exist in non-integral domains, but their properties differ significantly from those in integral domains.

PREREQUISITES
  • Understanding of commutative rings and their properties
  • Familiarity with the concept of ideals in ring theory
  • Knowledge of zero-divisors and their implications in algebra
  • Basic comprehension of integral domains and their characteristics
NEXT STEPS
  • Study the properties of zero-divisors in commutative rings
  • Explore the implications of prime ideals in non-integral domains
  • Learn about the structure of quotient rings, specifically \( R/J \)
  • Investigate the differences between prime elements and irreducible elements in ring theory
USEFUL FOR

Students and researchers in abstract algebra, particularly those focusing on ring theory and the properties of prime elements in various algebraic structures.

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On page 284 Dummit and Foote in their book Abstract Algebra define a prime element in an integral domain ... as follows:View attachment 5660My question is as follows:

What is the definition of a prime element in a ring that is not an integral domain ... does D&F's definition imply that prime elements cannot exist in a ring that is not an integral domain ... but why not ...?Can someone please clarify this situation ...

Peter
 
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The definition is aiming at the following theorem:

For a commutative ring $R$ and an ideal $J$:

$J$ is prime $\iff \ R/J$ is an integral domain.

The definition of prime element is the same for a mere commutative ring, but commutative rings with zero-divisors aren't so well-behaved, and primes aren't as "useful" there.
 

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