Exact embedding functors and left coherent rings

Abstract:

Let $R$ and $S$ be rings with unit. Suppose $P$ is a free $R$-module on $\beta$ generators, where $\beta$ is an infinite cardinal number not smaller than the cardinality of $R$, and $T$ is the ring of endomorphisms $\operatorname {End}{(_R}P)$. Theorem. If $R$ is left coherent and there exists an exact embedding functor $F:R - \operatorname {Mod} \to S - \operatorname {Mod}$, then $_SF{(R)_R}$ is a bimodule such that $F{(R)_R}$ is faithfully flat. Theorem. If $F:R - \operatorname {Mod} \to S - \operatorname {Mod}$ is an exact embedding functor, then $_R{P_T}$ is a bimodule such that $_RP$ is a projective generator (inducing an exact embedding Hom functor from $R - \operatorname {Mod}$ into $T - \operatorname {Mod}$,) and $_SF{(T)_T}$ is a bimodule such that $F{(T)_T}$ is faithfully flat (inducing an exact embedding tensor product functor $_SF(T){ \otimes _T}$ — from $T - \operatorname {Mod}$ into $S - \operatorname {Mod}$.) Theorem. There exists an exact embedding functor $R - \operatorname {Mod} \to S - \operatorname {Mod}$ iff there exists an $S$-module $N$ and a unit-preserving ring monomorphism $h:\operatorname {End}{(_R}P) \to \operatorname {End}{(_S}N)$ of their endomorphism rings, such that $h$ preserves and reflects exact pairs of endomorphisms.