Prime elements in a commutative ring with identity

Question

Prime numbers in $\mathbb{Z}$ are numbers where its only factors are 1 and itself.

Let R be a commutative ring with identity.

Definition of a prime element for commutative rings with identity: A nonzero nonunit p $\in$ R is called a prime element if whenever p $\vert$ ab in R, either p $\vert$ a or p $\vert$ b.

Does the same statement hold then, that an element r of R is prime if its only factors are 1 and itself? It seems to be true as if p = 7, 7 $\vert$ 1 or 7 $\vert$ 7.

Answer 1

Recall the following definition:

Let $R$ be a (unital) ring. An non-zero and non-unit element $x\in R$ is called irreducible if it is not the product of two non-units.

In an integral domain (no zero divisors) $R$, every prime element is automatically irreducible. However, in general irreducible elements need not be prime.

In many rings (such as unique factorization domains) irreducible elements coincide with prime elements.