Module theory MOC

Balanced product

A balanced product is a certain generalization of a bilinear map for a general module over a (noncommutative) ring 𝑅. Let 𝑀 be a right 𝑅-module, 𝑁 be a left 𝑅-module, and 𝐺 be an abelian group (ℤ-module). A map 𝜑 :𝑀 ×𝑁 →𝐺 is 𝑅-balanced iff for all 𝑚,𝑚′ ∈𝑀, 𝑛,𝑛′ ∈𝑁, 𝑟 ∈𝑅 module

1. 𝜑(𝑚,𝑛 +𝑛′) =𝜑(𝑚,𝑛) +𝜑(𝑚,𝑛′)
2. 𝜑(𝑚 +𝑚′,𝑛) =𝜑(𝑚,𝑛) +𝜑(𝑚′,𝑛)
3. 𝜑(𝑚 ⋅𝑟,𝑛) =𝜑(𝑚,𝑟 ⋅𝑛)

Together, ^B1 and ^B2 demand biadditivity. Just as bilinear maps are linear maps from the tensor product, 𝑅-balanced maps are homomorphisms from the Tensor product of modules over a noncommutative ring.

Examples

• Any ring 𝑅 may be regarded as an 𝑅-Bimodule, in which case the ring multiplication is balanced.

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