Let R be an associative ring
with 1, E a unitary right module, and (Fi)i∈I a family of unitary left modules. Let
f : E ⊗R TI Ft→∏(E ⊗RFi) be the canonical map. THEOREM. f is bijective
(surjective) for all families (Fi) iff E is finitely presented (finitely generated).
Theorem. If R is a Dedekind domain or is commutative artinian and every Fi is
flat, then f is injective. COROLLARY. If R is a Dedekind domain or is
commutative artinian, every Fi is flat and E ⊗RFi is reduced, then E ⊗R∏Ft is
reduced. THEOREM. If R is a Dedekind domain or is commutative artinian,
Ej is flat, f is injective for every Ej (e.g. Ef projective) and E is pure in
∏Ej, then f is injective. THEOREM. If R is a Dedekind domain and E is
flat then f is injective for E iff f is injective for Hom(F,E) for all modules
F. THEOREM. If R is a Dedekind domain and f is injective for E for all
families (Fi) then E is reduced. THEOREM. If R is commutative and f
is always injective then R must be artinian. The converse holds for serial
rings.
