Vol. 43, No. 1, 1972

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Distributing tensor product over direct product

Kenneth R. Goodearl

Vol. 43 (1972), No. 1, 107–110

This paper is an investigation of conditions on a module A under which the natural map

A ⊗ (ΠCα) → Π (A ⊗ Cα)

is an injection. The investigation leads to a theorem that a commutative von Neumann regular ring is self-injective if and only if the natural map

(ΠF α)⊗ (ΠG β) → Π(Fα ⊗ G β)

is an injection for all collections {Fα} and {Gβ} of free modules. An example is constructed of a commutative ring R for which the natural map

R[[s]]⊗R [[t]] → R[[s,t]]

is not an injection.

Mathematical Subject Classification
Primary: 16A50
Received: 28 July 1971
Published: 1 October 1972
Kenneth R. Goodearl
University of California, Santa Barbara
Santa Barbara CA
United States