Let $A$ be a unital, possibly noncommutative ring. Dischinger showed [1] that the following are equivalent:

1. For every $a \in A$, there exists $n \in \mathbb N$ such that $a^n A = a^{n+1} A$;

2. For every $a \in A$, there exists $n \in \mathbb N$ such that $A a^n = A a^{n+1}$;

3. Every cyclic right module $M \in Mod_A$ is co-hopfian (i.e. every injective endomorphism of $M$ is an isomorphism);

4. Every cyclic left module $M \in {}_A Mod$ is co-hopfian.

Such a ring $A$ is called strongly $\pi$-regular. Every strongly $\pi$-regular ring $A$ is $\pi$-regular in the sense that for every $a \in A$ there is some $n \in \mathbb N$ such that $a^n$ is a von Neumann regular element of $A$ (i.e. $a^n = a^n b a^n$ for some $b \in A$). Every von Neumann regular ring is strongly $\pi$-regular, and if $A$ is commutative, then $A$ is strongly $\pi$-regular iff $A/nil(A)$ is von Neumann regular.

Dischinger also showed that the following are equivalent:

1. Every finitely-generated right module $M \in Mod_A$ is co-hopfian;

2. Every finitely-generated left module $M \in {}_A Mod$ is co-hopfian;

3. Every finite-rank matrix ring over $A$ is strongly $\pi$-regular;

4. Every ring right (or left) Morita equivalent to $A$ is strongly $\pi$-regular.

Let's call such a ring $A$ very strongly $\pi$-regular. Every von Neumann regular ring is very strongly $\pi$-regular.

Questions:

1. Is there a direct ring-theoretic, rather than module-theoretic, characterization of very strongly $\pi$-regular rings which doesn't explicitly mention matrix rings?

2. What is an example of a very strongly $\pi$-regular ring $A$ such that $A$ is not von Neumann regular but the center $Z(A)$ is von Neumann regular?

[1] Dischinger, Friedrich. “Sur Les Anneaux Fortement π-Réguliers.” CR Acad. Sci. Paris Sér. AB 283, no. 8 (1976): 571–573. link.