Examples of rings having reducible primes

Question

In domains, all primes are at the same time irreducible. What happens when the ring is not a domain?

I need a practical example of this kind of ring. Thanks

Answer 1

The proof that primes are irreducible relies on the fact that $R$ has a $1$ and no zero-divisors. If $p=ab$ is a primes and $a,b$ are no units, then wLoG $p|a$, thus there is a $d$ such that $a=pd$. But then $p\cdot 1=p=ab=pdb$. We assume $p\neq0$, so we can cancel it and conclude that $db=1$ and $b$ is a unit. Hence $p$ is irreducible.

This fails if $R$ has zero-divisors. For example, in $\Bbb Z_6$ the element $2$ is the result of $4\cdot2$ and $4,2$ are not units. However, $2$ is a prime, for if $2$ does not divide $a$ and $b$, then $a,b\in\{1,3,5\}$, so $ab\in\{1,3,5\}$ and $p$ doesn't divide $ab$ either.