A ring $R$ is artinian if it is artinian as an $R$-module and $R$ is principal ideal ring if all its left and right ideals are principal. We notice that every artinian ring has artinian quotient rings (homomorphic images), ($R/I$), for any ideal $I$.
My problems are the following: I am looking for examples of rings either without any nonzero principal homomorphic images or without any artinian homomorphic images. That is:
(1) Are there rings without any nonzero principal quotients (no principal homomorphic images)?
(2) Are there rings without any nonzero artinian quotients (no artinian homomorphic images)?
My examples are the following:
(a) Let $D$ be a division ring and $V$ a countably infinite dimensional vector space over $D$. The ideal $I:=\{a\in \text{End}_D(V):\text{dim}_D(aV)<\infty\}$ is the only proper ideal of $\text{End}_D(V)$. I think that $\text{End}_D(V)/I$ is never artinian. Morever, $I$ is the only ideal of $\text{End}_D(V)$. So $\text{End}_D(V)$ is an example satisfying (2).
However, I do not have an example for (1). I would like to test as many examples as possible.
Your example works for both.
A left/right principal ideal ring is necessarily left/right Noetherian, and none of your example's nonzero quotients are Noetherian or Artinian.
This is because a von Neumann regular ring is Noetherian if and only if it is semisimple. Your example (and its nontrivial quotient) are both nonArtinian von Neumann regular rings.