I am trying to tie together the various elements of the definition of a prime element in a ring/integral domain. [why integral domain and not ring??]

On page 284 Dummit and foote define a prime element as follows:

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"Let R be an integral domain.

The non-zero element p R is called prime if the ideal (p) generated by p is a prime ideal., a non-zero element p is a prime if it is not a unit and whenever p | ab for any a, b R, then either p | a or p | b."In other words

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I need help to understand how the definition of a prime in terms of a prime ideal ties up with the definition given after "In other words".

In particular why does (p) being a prime ideal imply that p cannot be a unit.

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Further if you check D&F's definition of a prime ideal it seems to specify things in terms of inclusion rather than p dividing elements.

Specifically the definition of a prime ideal on page 255 is as follows:

"Assume R is commutative. An ideal P is called a prime ideal if P R and whenever the product ab of two elements a, b R is an element of P, then at least one of a and b is an element of P.

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Can someone please clarify the above for me. In particular why is the definition of prime ideal couched in terms of set inclusion while the definition of a prime element is couched in terms of divisors?

Peter