Ring extensions and essential monomorphisms

Abstract

We study pairs of rings R ⊂ S R \subset S such that Hom R ⁡ ( S , − ) : R − Mod → S − Mod \operatorname {Hom}_R(S, - ):R - \operatorname {Mod} \to S - \operatorname {Mod} preserves essential monomorphisms. We obtain a complete characterization of such a pair in case S is a torsion-free algebra over a Noetherian domain R ≠ Q u o t ( R ) R \ne \mathrm {Quot}(R) ; S is then a left ideally finite R-algebra. The rings R such that every ring extension R ⊂ S R \subset S satisfies the above condition are subdirect sums of certain Artinian rings. Furthermore, we study a generalization of trivial ring extensions and show that the center of a semi-Artinian ring is again semi-Artinian.