Prime Elements in Non-Integral Domains?

In summary, on page 284 of their book Abstract Algebra, Dummit and Foote define a prime element in an integral domain as follows: for a commutative ring R and an ideal J, J is prime if and only if R/J is an integral domain. This definition can also be applied to commutative rings with zero-divisors, but the usefulness of prime elements is limited in these types of rings.
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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.
 

FAQ: Prime Elements in Non-Integral Domains?

What is a prime element in a ring?

A prime element in a ring is an element that cannot be written as a product of two non-units. In other words, it is an element that has no non-trivial factors.

How do you determine if an element is prime in a ring?

To determine if an element is prime in a ring, you can use the definition and check if the element has any non-trivial factors. Alternatively, you can also use the concept of irreducibility, where an element is prime if and only if it is not a unit and cannot be written as a product of two non-units.

What is the difference between a prime element and an irreducible element?

A prime element is an element that cannot be written as a product of two non-units, while an irreducible element is an element that cannot be factored into smaller non-units. In other words, every prime element is irreducible, but not every irreducible element is necessarily prime.

Can a ring have more than one prime element?

Yes, a ring can have more than one prime element. In fact, most rings have multiple prime elements. However, there are some rings, such as integral domains, where every prime element is also irreducible, so there can only be one prime element in these rings.

Are prime elements the same as prime numbers?

No, prime elements in a ring are not the same as prime numbers. While prime numbers are used in the context of integers, prime elements are used in the context of rings. However, there are some similarities, such as both being irreducible and having no non-trivial factors.

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