# Is every algebra a coalgebra?

I was looking to study the topic of coalgebras and few examples were given. In particular, I didn't see anywhere this example: let $A$ be any $\mathbb{K}$-algebra, in particular it is a $\mathbb{K}$ vector space, let $\{a_i\}_{i \in I}$ be a base, then $A$ is a coalgebra with coproduct $\Delta(a_i) = a_i \otimes a_i$ and $\epsilon = 1$. I have seen this example but applied only to very specific algebras (like polynomials). It seems odd that is never said that any algebra is canonically a coalgebra.

• That is because the map is not a homomorphism of algebras in general. To see this consider for example the algebra $k[x]/(x^2-2x)$ and the obvious basis (and see what happens when you square either side). Nov 3 '16 at 22:20
• @TobiasKildetoft OP never said she wants $A$ to be a bialgebra, though. Nov 3 '16 at 22:24
• @PedroTamaroff Ohh, good point (and of course the example I gave is actually a Hopf algebra with the given comultiplication as long as the basis is chosen differently). Nov 3 '16 at 22:25
• Right, I was thinking of bialgebra. I have never seen coalgebras studied on their own (even though reps of algebraic groups are technically comodules for a coalgebra, the fact that it is really a Hopf algebra is needed in most places). Nov 3 '16 at 22:29
• @Eli: the example is correct but no use is made of the algebra structure on $A$ so why mention it? Oct 3 '17 at 3:09

Your observation is correct and it is in fact valid in an even more general setting. It actually does not have to be a $k$-algebra, it works for any set:
Let $S$ be a non-empty set; $kS$ is the $k$ vector space with basis $S$. Then $kS$ is a coalgebra with comultiplication $\Delta$ and counit $\epsilon$ defined by $\Delta(s)=s\otimes s$, $\ \epsilon(s)=1$ for any $s\in S$.
This shows in particular, that any $k$-vector space can be endowed with a coalgebra structure (as has already been mentioned in a comment above by user Ender Wiggins).