A non-trivial unitary ring $R$ (not necessarily commutative) is called a $\Pi$-ring [2] [3], if for any finitely generated $R$-module $M$ and any $R$-module $N$, if there exists an injective $R$-linear map $i:N\to M$ and a surjective $R$-linear map $f:N\to M$, then $f$ is also injective.
In [2] [3] it's proved that a $\Pi$-ring satisfies the strong rank condition, that is, if there exists an injective $R$-linear map $i:R^n\to R^m$, then $n\leq m$. (In fact, suppose $n>m$, then there exists a surjective $R$-linear map $f:R^n\to R^m$ which is not injective, contradicts with the $\Pi$-ring condition.)
In [1] [2] [3] it's proved that a Noetherian ring (not necessarily commutative), as well as a commutative ring, is a $\Pi$-ring. On the other hand, by other method it's known that such ring also satisfies the strong rank condition. This makes me wonder if the $\Pi$-ring is equivalent to the strong rank condition. Is it true? If not, are there counterexamples?
References:
- Orzech, Morris. Onto endomorphisms are isomorphisms. Amer. Math. Monthly 78 (1971), 357--362.
- Djoković, D. Ž. Epimorphisms of modules which must be isomorphisms. Canad. Math. Bull. 16 (1973), 513--515.
- Ribenboim, Paulo. Épimorphismes de modules qui sont nécessairement des isomorphismes. (French) Séminaire P. Dubreil, M.-L. Dubreil-Jacotin, L. Lesieur et C. Pisot (24ème année: 1970/71), Algèbre et théorie des nombres, Fasc. 2, Exp. No. 19, 5 pp. Éditions de l'Académie de la République Socialiste de Roumanie, Bucharest, 1971.
Related questions:
Is Orzech's generalization of the surjective-endomorphism-is-injective theorem correct?
A Ring with Rank Condition that doesn't satisfy Strong Rank Condition
