Homotopy Type Theory bimodule > history (Rev #2)

Definiton

Let $R$ and $S$ be commutative rings and let $M$ be a left $R$-module and a right $S$-module, with a left multiplicative bilinear $R$-action $\alpha_l:R \times A \to A$ and a right multiplicative bilinear $S$-action $\alpha_r:A \times S \to A$. $M$ is a $R$-$S$-bimodule if

For a commutative ring $R$, a $R$-$R$-bimodule is also called a $R$-bimodule.

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
• module
• algebra (module theory)
• ideal (ring theory)
• vector space?