Vol. 49, No. 2, 1973

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ISSN: 0030-8730
Tensor and direct products

Cary Webb

Vol. 49 (1973), No. 2, 579–594
Abstract

Let R be an associative ring with 1, E a unitary right module, and (Fi)iI 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.

Mathematical Subject Classification
Primary: 16A62
Milestones
Received: 21 August 1972
Published: 1 December 1973
Authors
Cary Webb