nLab quotient bimodule

Idea

The quotient of a bimodule by a subbimodule.

Definition

Given rings $R$ and $S$, a $R$-$S$-bimodule $B$, and a sub-$R$-$S$-bimodule $I$ with a $R$-$S$-bimodule monomorphism $i:I \hookrightarrow B$, the quotient of $B$ by $I$ is the initial $R$-$S$-bimodule $B/I$ with a $R$-$S$-bimodule homomorphism $h:B \to B/I$ such that for every element $a \in I$, $h(i(a)) = 0$: for any other $R$-$S$-bimodule $A$ with a $R$-$S$-bimodule homomorphism $k:B \to A$ such that for every element $a \in I$, $k(i(a)) = 0_A$, there is a unique $R$-$S$-bimodule homomorphism $l:B/I \to A$ such that $l \circ h = k$.

Related concepts

• quotient group

• quotient module

• quotient ring