
bialgebra, coideal
 Hi! These are my problems, thanks:
1)
I can't figure out what bialgebra is. In my book (Montgomery "Hopf Algebras and Their Actions on Rings") it is said:
"A kspace 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)
In Wikipedia: Bialgebra  Wikipedia, the free encyclopedia.
So I can't understand why this is the same thing. Why is there a twist map in the first diagram?
2)
"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 kspace 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?