Definitions of prime element:
$(1)$ We say $p$ is prime if $p|ab$ it implies $p|a$ or $p|b$ (I don't need definition of unity here)
$(2)$ We say $p$ is prime if $p=ab$ it implies $p|a$ or $p|b$ (I don't need definition of unity here)
Are these two definitions equivalent?
Note: $p = ab$ it may not imply $p|ab$ using definition $(1)$ (as much as I can see because there is no unity)
Motivation: ring without unity with an prime element $p$ such that $ab=p$ but $p$ does not divide $a$ nor it divides $b$ (if possible)
An example where (1) and (2) are not equivalent is the Rng $2\mathbb{Z}$.
If we take definition (1), then 2 is not prime since if $a=b=6$, we have $ab=36$ and $2|36$ since $36=2\times 18$. However 2 does not divide 6 in $2\mathbb{Z}$ as we cannot write $6=xy$ where $x,y\in 2\mathbb{Z}$.
On the other hand 2 is prime according to (2) since the condition holds vacuously.