I can't figure out what bialgebra is. In my book (Montgomery "Hopf Algebras and Their Actions on Rings") it is said:
"A k-space B is a bialgebra if (B, m, u) - is an algebra, (B, d, e) - is a coalgebra, and either of the following (equivalent) conditions holds:
1) d and e are algebra morhisms,
2) m and u are coalgebra morphisms.
(m - multiplication, u - unit, d - comultiplication, e - counit)
So I can't understand why this is the same thing. Why is there a twist map in the first diagram?
"A subspace I of coalgebra C is a coideal if d(I) lies in IxC+CxI (x - tensor product), and if e(I)=0.
It is easy to check that if I is a coideal, then the k-space C/I is a coalgebra with a comultiplication induced from d..."
I think i almost get it, but i can't write it down strictly. How to prove it correctly?