nLab biideal

Given a bigebra $B$ over a field $k$ with comultplication $\Delta$, a $k$-vector subspace $I$ is a biideal if it is two sided ideal (i.e. for all $b\in B$, $b I\subseteq I$ and $I b\subseteq I$) and a coideal, i.e. $\Delta(I)\subseteq I\otimes B + B\otimes I$.

Quotient of a bigebra $B$ by a biideal $I$ is itself inheriting a canonical structure of a bigebra by taking representatives both for multiplication and for comultiplication of classes. This is the quotient bigebra.

A Hopf ideal is a biideal in a Hopf algebra which is invariant (as a set) under the antipode map. A quotient bigebra of a Hopf algebra is a Hopf algebra iff the biideal is in fact a Hopf ideal.