Let $A$ be an $R$-algebra.
Suppose $A$ has a $R$-coalgebra structure compatible with the algebra structure. (I.e. there is a comultiplication map $\Delta$ and counit map $\epsilon$ compatible with the multiplication map and unit map of $A$.)
Then I am wondering whether $A$ can have another $R$-coalgebra structure other than $(\Delta, \epsilon)$?