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Tensor Products and Flat Modules

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Algebra

Part of the book series: Die Grundlehren der mathematischen Wissenschaften ((GL,volume 190))

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Abstract

Let R be a ring, M a right and N a left R-module. As usual, M × N denotes cartesian product. If G is an abelian group, and g: M× N → G is a mapping of sets, then for each y ∈ N there is a mapping g y : M → G, defined by the formula g y (α) = g (α, y) ∀ α ∈ M. Symmetrically, if x ∈ M, then g x : N → G is defined by the formula g x (b) = g (x, b) ∀b ∈ M. A mapping g:M×N → G is bilinear in case g y : M→G and g x : N → G are group homomorphisms ∀ x ∈ M,y ∈ N. A balanced map g:M×N →G is a bilinear map such that

$$g\left( {xr,y} \right) = g\left( {x,ry} \right)$$

∀ x ∈M, y ∈ N, r∈R.

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    Carl Faith

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Faith, C. (1973). Tensor Products and Flat Modules. In: Algebra. Die Grundlehren der mathematischen Wissenschaften, vol 190. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80634-6_13

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