Irreducible elements for a commutative ring that is not an integral domain

Solution 1:

In an integral domain, you have the following four equivalent definitions for a nonzero nonunit $a$ to be irreducible.

1. $a = bc \Rightarrow (a) = (b)$ or $(a) = (c)$.
2. $a = bc \Rightarrow a$ is a unit multiple of $b$ or $c$.
3. $(a)$ is maximal among the proper principal ideals.
4. $a = bc \Rightarrow b$ or $c$ is a unit.

However, in commutative rings in general, we have (4) $\Rightarrow$ (3) $\Rightarrow$ (2) $\Rightarrow$ (1), and none of the implications reverse. The literature for factorization in commutative rings with zero divisors thus has four different non-equivalent definitions of "irreducible". (The above statements define "irreducible", "strongly irreducible", "m-irreducible", and "very strongly irreducible", respectively.) See Factorization in Commutative Rings with Zero Divisors by Anderson and Valdes-Leon for more information.