Let $R$ be a $k$-algebra, $k$ a field, and $0\neq I \neq R$ a two-sided ideal of $R$.
Denote by P a property of rings. One says that $R$ is 'just P', if $R$ does not satisfy property P but $R/I$ satisfies property P.
Example: P= finite dimensional over $k$, see, for example, this paper.
What is known about P= every injective $k$-algebra endomorphism is surjective, hence bijective. In other words, P= every monomorphism is an epimorphism, hence an automorphism.
In particular, I wonder if this may be helpful for the Dixmier Conjecture, namely, $R$ is the first Weyl algebra over $k=\mathbb{C}$. In this case, $R$ is simple, so every endomorphism is a monomorphism. What can be said about the quotient rings of $R$?
Truly, I first thought about the following situation: $R=\mathbb{C}[x,y]$. Assume that $p,q \in \mathbb{C}[x,y]$ satisfy $\operatorname{Jac}(p,q)=1$. I wonder if it may be interesting to consider $I$ such that $p,q \notin I$ (I guess that further restrictions are required, except for the obvious one $\bar{p} \neq \bar{0}, \bar{q} \neq \bar{0}$).
First we should ask when $\frac{\mathbb{C}[x,y]}{I} = \mathbb{C}[\bar{p},\bar{q}]$? if this happens then $x+I,y+I \in \mathbb{C}[\bar{p},\bar{q}]$. Then $x+I=u(\bar{p},\bar{q})=u(p,q)+I$ and $y+I=u(p,q)+I$, for some $u,v \in \mathbb{C}[s,t]$. Therefore, $x-u(p,q) \in I$ and $y-v(p,q) \in I$. Perhaps this is relevant somehow to the Jacobian Conjecture, I do not know yet; maybe another special case can be obtained for the JC (taking just one ideal $I$ does not help much, for example $I=(x-p,y-q)$).
Thank you very much!