I have seen many proofs about a prime element is irreducible, but up to now I am thinking whether this result is true for any ring.
Recently, I got this proof:
Suppose that $a$ is prime, and that $a = bc$. Then certainly $a\mid bc$, so by definition of prime, $a\mid b$ or $a\mid c$, say $a \mid b$. If $b = ad$ then $b = bcd$, so $cd = 1$ and therefore $c$ is a unit. (Note that $b$ cannot be $0$,for if so, $a = bc = 0$, which is not possible since $a$ is prime.) Similarly, if $a\mid c$ with $c = ad$ then $c = bcd$, so $bd = 1$ and $b$ is a unit. Therefore $a$ is irreducible.
I think with the above proof we do not need the ring to be an integral domain. If this is the case then I will stop doubting, else, I am still in it.
Can somebody help me out?