What kind of algebraic structure is this

Question

I know that a commutative ring with an additional scalar multiplication on it is called an associative algebra. If the ring also has a 1 it is called a unital algebra. What would you call a field with an additional scalar multiplication(from a different field) on it?

Answer 1

If $k$ is a commutative ring, then a commutative $k$-algebra $A$ is the same as a homomorphism of rings $k \to A$. If $k,A$ are fields, then $k \to A$ is injective. Therefore this is just a field extension.

Answer 2

An associative algebra over a commutative ring is just an associative ring $A$ which has some scalar multiplication from a commutative ring $R$ which satisfies some axioms.

If both $A$ and $R$ are fields, I'd still call $A$ an $R$-algebra. You might improve by calling it an $R$ division algebra.