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 $𝜑 : 𝑀 \times 𝑁 \to 𝐺$ is $𝑅$ -balanced iff for all $𝑚, 𝑚' \in 𝑀$ , $𝑛, 𝑛' \in 𝑁$ , $𝑟 \in 𝑅$ #m/def/module

$𝜑 (𝑚, 𝑛 + 𝑛') = 𝜑 (𝑚, 𝑛) + 𝜑 (𝑚, 𝑛')$
$𝜑 (𝑚 + 𝑚', 𝑛) = 𝜑 (𝑚, 𝑛) + 𝜑 (𝑚', 𝑛)$
$𝜑 (𝑚 \cdot 𝑟, 𝑛) = 𝜑 (𝑚, 𝑟 \cdot 𝑛)$

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

Examples

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

#state/tidy | #lang/en | #SemBr