Definition:Prime Element of Ring

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Let $R$ be a commutative ring.

Let $p \in R \setminus \left\{{0}\right\}$ be any non-zero element of $R$.

Then $p$ is a prime element of $R$ if and only if:

$(1): \quad$ $p$ is not a unit of $R$
$(2): \quad$ whenever $a, b \in R$ such that $p$ divides $a b$, then either $p$ divides $a$ or $p$ divides $b$.

Also see

Special Cases