Definition:Prime Element of Ring

Definition

Let $R$ be a commutative ring.

Let $p \in R \setminus \set 0$ 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

• Definition:Prime Number

Generalizations

• Definition:Prime Ideal of Ring, as shown in Prime Element iff Generates Principal Prime Ideal