Given a bigebra BB over a field kk with comultplication Δ\Delta, a kk-vector subspace II is a biideal if it is two sided ideal (i.e. for all bBb\in B, bIIb I\subseteq I and IbII b\subseteq I) and a coideal, i.e. Δ(I)IB+BI\Delta(I)\subseteq I\otimes B + B\otimes I.

Quotient of a bigebra BB by a biideal II 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.