Difference between rings and algebras

Question

It is clear that, if we forget scalar multiplication, associative algebras reduce to rings. It the resulting ring is unital however, one can recover scalar multiplication $ka:=(k1)a$, for all $k\in\mathbb{F}$, the field where the algebra is defined, and $a$ in the algebra. So what is the difference between unital rings and associative unital algebras?

Answer 1

Notice that to define $ka=(k1)\cdot a$, you have to already know what $k1$ is. If all you have is a ring, then you don't know what $k1$ is yet. The same (unital) ring can have different $\mathbb{F}$-algebra structures, since $k1$ could be defined differently in them.

Answer 2

The main difference is this: Algebras have a scalar multiplication defined, rings don't. The fact that the multiplication can be recovered to make some rings into algebras in a canonical way is more or less irrelevant.