Prime Elements in Non-Integral Domains?

[Math Amateur]

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

 

[Deveno]

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.