  The grouplike elements of a coalgebra over a field are known to be linearly
independent over said field. Here we prove three variants of this result. One
is a generalization to coalgebras over a commutative ring (in which case the
linear independence has to be replaced by a weaker statement). Another is a
stronger statement that holds (un-der stronger assumptions) in a commutative
bialgebra. The last variant is a linear independence result for characters (as
opposed to grouplike elements) of a bialgebra.
