TITLE: Irreducible/prime/indivisible elements
QUESTION [10 upvotes]: in what follows all the rings are commutative, nontrivial, with unit.
Recall the following definitions:
1) $\pi\in A$ is prime if $(\pi)$ is a nonzero prime ideal
2) $\pi\in A$ is irreducible if $\pi$ is nonzero, non invertible, and for all $a,b\in A$, $\pi=ab$ implies that $a$ or $b$ is a unit.
3) $\pi\in A$ is indivisible if $\pi$ is nonzero, non invertible,  and  for all $a,b\in A$, $\pi=ab$ implies that $\pi\mid a$ or $\pi\mid b$.
The following facts are known:

if $A$ is a domain, prime$\Rightarrow$ irreducible.

This is not true anymore if $A$ has zero divisor (e.g. $A=\mathbb{Z}/6\mathbb{Z}$: $A$ has prime elements, but no irreducible elements)

irreducible $\Rightarrow$ indivisible
there exist indivisible elements which are not irreducible: $A=\mathbb{Z}/6\mathbb{Z}$, $\pi=3$ . However, this one is prime.
if $A$ is a domain, irreducible $\iff$ indivisible
if $A$ is noetherian, $A$ has indivisible elements
if $A$ is a noetherian domain which is not a field, $A$ has irreducible elements.

After this lengthy introduction, let me ask the following questions:

Q1: can we find an example of an indivisible element which is neither prime or irreducible ? If possible, I would like $A$ to be noetherian or, even better, finite.
Q2: can we find an example of a noetherian ring which is not a field, which has no prime elements AND no irreducible elements ? (so necessarily, $A$ has zero divisors)
Q3: can we find an example of a noetherian domain which is not a field which has no prime elements ?
Q4: if the answer to Q3 is NO, can we find an example of a  domain which has irreducible elements, but which has no prime elements ?

Thanks for your time.
Greg

REPLY [5 votes]: It looks like these questions have been taken care of, but I just thought I would mention a few references and talk about other similar concepts in between the definitions you provide here!  I find this kind of thing very interesting, so I am always excited to see others interested in it as well.
If you are interested in even more of these kinds of irreducible properties, I would suggest taking a look at the paper by D.D. Anderson and S. Valdez-Leon http://projecteuclid.org/euclid.rmjm/1181072068
Many of these concepts of irreducible/indivisible diverge with zero-divisors present.  It may be a bit confusing because what you have called indivisible is what is called irreducible in this paper.  Your definition of irreducible is very strongly irreducible.  In fact there are two other notions that are in between these two concepts.  $\pi$ is m-irreducible if it is maximal among principal ideals. $\pi$ is strongly irreducible if $\pi=bc$ implies $\pi \approx b$ or $\pi \approx c$ where $a\approx b$ means there is a unit $\lambda$ such that $a=\lambda b$.
Then for non-zero elements you have very strongly irreducible $\Rightarrow$ m-irreducible $\Rightarrow$ strongly irreducible $\Rightarrow$ irreducible.  Lastly, prime $\Rightarrow$ irreducible (your indivisible) but none of the others.
If you look at example 2.3 in the article I linked, they provide the example $F[X,Y,Z]/(X-XYZ)$ where $F$ is a field.  Then $x,y,z$ are the image of $X,Y,Z$, respectively.  Then $x$ is prime, and hence irreducible (indivisible in your terms) but not even strongly irreducible and hence neither m-irreducible nor very strongly irreducible.  Other examples showing these are distinct in rings with zero-divisors are provided in this article following Theorem 2.12.
Regarding the last few questions I would point you towards (ON INTEGRAL DOMAINS WITH NO ATOMS) which are cleverly called "anti-matter domains" https://www.ndsu.edu/pubweb/~coykenda/paper13.pdf