Definition:Prime Element of Ring

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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

• Definition:Prime Number
• Definition:Prime Ideal of Ring
• Prime Element iff Generates Principal Prime Ideal

• Results about prime elements of rings can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): prime ideal