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What it says

Prime Divisibility: $ p \mid ab \Rightarrow p \mid a \lor p \mid b $

For any elements $a$ and $b$ in a commutative monoid with zero, if $p$ is a prime element and $p \mid ab$, then $p \mid a$ or $p \mid b$.

What it says

Prime Divides Product iff Divides an Element in the List

In a commutative monoid with zero, a prime element $p$ divides the product of a list $L$ of elements if and only if there exists an element $a$ in $L$ such that $p$ divides $a$. This equivalence characterizes the prime property in terms of divisibility of products.

What it says

Prime Divisibility Lemma: $ a \mid p \cdot b $ implies $ p \mid a $ or $ a \mid b $

In a commutative monoid with zero, if $p$ is a prime element and an element $a$ divides the product $p \cdot b$, then $p$ divides $a$ or $a$ divides $b$. This follows from the definition of prime elements, which are non-zero, non-unit elements satisfying the property that divisibility of a product implies divisibility of one of the factors.

What it says

Prime Divides Product Implies Divides a Factor

If a prime number $p$ divides the product of two integers $m$ and $n$, then $p$ divides $m$ or $p$ divides $n$.

What it says

Prime Divides Product if and Only if Divides a Factor

If $p$ is a prime number and $p$ divides the product $m imes n$ of natural numbers $m$ and $n$, then $p$ divides $m$ or $p$ divides $n$, and conversely.