# Examples for regular ideals in commutative rings

Rings are commutative with 1.

Definition. A regular ideal refers to an ideal containing a non-zero-divisor.

I can easily build ideals which are not regular. but what about regular ideals? I know trivial ones:
1- A ring $$R$$ has at least one regular ideal, $$R$$ itself. 2- In an integral domain nonzero ideal is regular.

How can I build other (non-trivial) ones?

thank you

• This isn't my field, but I think one reason this hasn't been answered yet is that the question is a little unclear. Are you just looking for a handful of examples of regular ideals that don't fit in the given categories? Are you looking for a way to build a proper regular ideal out of any ring? Or are you perhaps looking for a way to generate all regular ideals? – Theo Bendit Oct 28 '18 at 9:19
• looking for a handful of examples of regular ideals that don't fit in the given categories – 13571 Oct 31 '18 at 6:23
• It all depends on the structure of zero divisors in the ring. Sometimes there is really not much to say about regular ideals. Often rings are classified by descriptions of their regular ideals. A few pervasive concepts here are: a Marot ring, in which every regular ideal is generated by regular elements; an additively regular ring, in which the coset $a + rR$ always contains a regular element when $r$ is regular; a ring with Property (A), in which f.g. ideals are regular iff their annihilator is trivial. Noetherian rings are, very importantly, all of these things. – Badam Baplan Oct 31 '18 at 13:42
• In $0$-dimensional rings, regular elements are units, and therefore the only regular ideal is $R$ itself. – Badam Baplan Oct 31 '18 at 13:42