19

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?

Bill Dubuque
  • 282,220
Hassan Muhammad
  • 4,522
  • 25
    The proof is incorrect if the ring is not a domain because $b=bcd$ does not imply $1=cd$, even if you assume $b\neq 0$. – Georges Elencwajg Oct 03 '11 at 09:49
  • If I get you Georges, primes are irreducible only in a domain. – Hassan Muhammad Oct 03 '11 at 09:52
  • 11
    It is worse than that, Hassan: I don't even think that the notion "irreducible" is a good one in rings with zero-divisors. On the other hand the notion of prime element, with the definition you used in your post, seems to me quite reasonable, even in a ring which isn't a domain. Of course this is somewhat subjective, but I have the feeling that I'm expressing a rather widely held consensus among algebraists. – Georges Elencwajg Oct 03 '11 at 10:48
  • Notes, In ring $\mathbb R \times \mathbb R$, we have $(b,0) = (b,0)(1,c)(1,d)$, obviously $(1,c)(1,d) \ne \mathbf 1$ – Zang MingJie Mar 07 '19 at 08:11
  • 1
    $b=bcd \Rightarrow b(cd-1) = 0 \Rightarrow b=0 \text{ or } cd=1$ Here requires $R$ is a domain – Zang MingJie Mar 07 '19 at 08:35
  • The statement holds only for an ID with unity, and not all ID's. Is that a correct conclusion? – utkarshk5 Jan 11 '21 at 10:13
  • related: https://math.stackexchange.com/q/1149078/173147 – glS Aug 06 '21 at 15:23

3 Answers3

26

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

Bill Dubuque
  • 282,220
13

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.

CPM
  • 4,003
  • 18
  • 23
0

This answer is just to give an easy way to generate examples of rings that contains a prime element that is not irreducible.

Let $R=D\times R^\prime$ where $D$ is a domain (by definition an integral domain is non-zero) and $R^\prime$ is a commutative unital non-zero ring. Then we claim that $(0,1)=(0,1)^2$ is a prime element that is not irreducible. The idea generates from the classical example $\mathbb{Z}/6\mathbb{Z}\cong\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$, $\overline{2}=\overline{2}\cdot \overline{4}$ is the same as $(0,-1)=(0,-1)\cdot (0,1)$, and $\overline{2}$ is prime but not irreducible.

Proof: To show that $(0,1)$ is prime it suffices to show $(0,1)\neq 0_R$ and $R/(0,1)R$ is a domain. Both of them are trivial by the construction of $R$. In particular, $(0,1)\notin R^\times$. So $(0,1)=(0,1)^2$ is not irreducible.

Z Wu
  • 2,099