Confusing with definition of (co)algebra

Question

I began to read the book « Hopf algebras » by Sweedler. He gave the definitions of algebra and coalgebra using commutative diagrams. These diagrams just show (co)associativity and (co)unitary property. As I understand, algebras need distributivity. Why there is not such property in the definitions?

Thanks!

Answer 1

I'll give you an example, a ring ($\mathbb Z−$algebra) is an abelian group $(\mathbb Z-$ module) $R$ together with a map $μ:R⊗_{\mathbb Z} R→R$ satisfying the associativity and unit conditions.

Define $μ(r⊗s)=r∗s$.

The reason we don't need distributivity is that

$$μ(a⊗(b+c))=a∗(b+c)$$

but

$$a⊗(b+c)=a⊗b+a⊗c$$

so

$$μ(a⊗(b+c))=μ(a⊗b+a⊗c)=a∗b+a∗c$$

which means that

$$ a*(b+c) = a∗b+a∗c $$

Similar argument shows that distributivity of multiplication from the right in $R$ holds aswell, so distributivity is already guaranteed by the fact that the product $⊗$ is distributive. Working with algebras over $\mathbb Z$ is simpler than over general rings so I'll leave it up to you to prove that distributivity is guaranteed in those cases aswell.