Homotopy Type Theory bimodule > history (Rev #5)

Definiton

A single action ring

Let $R$ be a ring. An $R$-bimodule is an abelian group $B$ with a trilinear multiplicative $R$-biaction $(-)(-)(-):R \times B \times R \to B$.

Two different action rings

Let $R$ and $S$ be rings. A $R$-$S$-bimodule is an abelian group $B$ with a trilinear multiplicative $R$-$S$-biaction $(-)(-)(-):R \times B \times S \to B$.

Properties

• Every abelian group is a $\mathbb{Z}$-$\mathbb{Z}$-bimodule.
• Every left $R$-module is a $R$-$\mathbb{Z}$-bimodule.
• Every right $R$-module is a $\mathbb{Z}$-$R$-bimodule.

See also

• abelian group
• trilinear function
• biaction
• module
• algebra (module theory)
• bimodule homomorphism
• ideal (ring theory)
• vector space?