Hopf ideals and Hopf subalgebras

Let $k$ be a field and let $H=(H,m,u,\mathrm{\Delta },\epsilon ,S)$ be a Hopf $k$-algebra as in Hopf algebras and antipodes, with $S$ satisfying both convolution antipode equations. A subbialgebra of $H$ is a $k$-subspace $K\subseteq H$ equipped with a bialgebra structure as in Bialgebras and bialgebra maps for which the inclusion $K\hookrightarrow H$ is a bialgebra map. A Hopf subalgebra is a subbialgebra $K$ with $S(K)\subseteq K$. Under AC, meaning every family of nonempty sets has a choice function (The Axiom of Choice), the tensor-injection lemma Tensoring injections and the kernel of a tensor product of quotient maps over a field identifies $K\otimes K$ with its image in $H\otimes H$; in that shorthand, the subbialgebra condition says exactly that $K$ is a unital subalgebra, ${\mathrm{\Delta }}_H(K)\subseteq K\otimes K$, and the counit is restricted from $H$.

A Hopf ideal is a two-sided ideal in the sense of Left, right and two-sided ideals, $I⊴H$, satisfying ${\epsilon }_H(I)=0$, $S(I)\subseteq I$, and the coideal condition from Subcoalgebras, coideals, and quotient coalgebras: ${\mathrm{\Delta }}_H(I)\subseteq \mathrm{im}(I\otimes H\to H\otimes H)+\mathrm{im}(H\otimes I\to H\otimes H)$. Under AC, the tensor-injection lemma identifies these tensor products with subspaces of $H\otimes H$, giving the usual shorthand ${\mathrm{\Delta }}_H(I)\subseteq I\otimes H+H\otimes I$. The two-sided ideal condition gives the unital quotient algebra $H\mathrm{/}I$; ${\epsilon }_H(I)=0$ makes its counit descend, and the remaining conditions are precisely those needed for the coproduct and antipode to descend. A subbialgebra is not called a Hopf subalgebra unless the antipode restricts to it.