# Principal Ideal Ring which is not Integral

In Atiyah & McDonald: Commutative Algebra the Principal Ideal Domain is a principal ideal ring which is also an integral domain.

I tried but couldn't find examples of commutative rings with identity that have the property that every ideal is generated by a single element but are not integral. Any suggestions ?

I believe that the fact that the set of all zero-divisors is not closed under "$+$" (in $\mathbb{Z_6}$ for example $3+2=5$) is making the search of such example difficult.

• $\mathbb Z_4$ not integral domain and has one proper ideal $\{0,2\}$ generated by $2$ – Mustafa Jan 15 '18 at 11:19
• Ironically, $\Bbb Z_6$ is such a ring. In general, if $R$ is a PID, then every quotient of $R$ is a PIR. – Crostul Jan 15 '18 at 11:19
• in general, all rings $\mathbb Z_n$ where $n$ is not prime – Mustafa Jan 15 '18 at 11:22

## 1 Answer

In general, if $R$ is a PID, then every quotient of $R$ is a PIR.

Ironically, $\Bbb Z_6$ is such a ring, because it is $\Bbb Z / (6)$.

• Everybody leapt on the 'quotient' idea which is of course excellent, but it's really a shame nobody mentioned the product of two PIDs is a PIR that isn't a domain. Of course, the example given here is also the product of two fields, so I think it would make a great addition! Regards – rschwieb Jan 15 '18 at 14:16