Is any prime element irreducible?
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?
Notice that your proof assumes that $\rm\: b\ne 0\ \Rightarrow\ b\:$ is cancellable, so it fails if $\rm\:b\:$ is a zero-divisor. Factorization theory is more complicated in non-domains. Basic notions such as associate and irreducible bifurcate into a few inequivalent notions. See for example
When are Associates Unit Multiples?
D.D. Anderson, M. Axtell, S.J. Forman, and Joe Stickles.
Rocky Mountain J. Math. Volume 34, Number 3 (2004), 811-828.
Factorization in Commutative Rings with Zero-divisors.
D.D. Anderson, Silvia Valdes-Leon.
Rocky Mountain J. Math. Volume 28, Number 2 (1996), 439-480
If you choose the definition of $a$ is irreducible if $a=bc$ implies that $(a)=(b)$ or $(a)=(c)$ then it is true actually. For instance the proof is as follows:
Let $p\in R$ be a non-zero, non-unit. Suppose $p=bc$. We clearly have $b\mid p$ and $c \mid p$ since $b$ and $c$ are factors of $p$. On the other hand, $1\cdot p=bc$ implies that $p \mid bc$ and $p$ being prime implies $p \mid b$ or $p \mid c$. Thus $(p)=(b)$ or $(p)=(c)$ showing $p$ is irreducible by this definition.
Unfortunately, prime is quite different than any of the other possible definitions of irreducible when zero-divisors are present that I am familiar with. The definition above is the weakest choice for irreducible that I am aware of.