I am reading about modules and some days ago I've worked on some exercises related to $k$-algebras. The definition I've seen of $k$-algebra is that it is a field $k$ and a ring $A$ together with a ring morphism $\rho:k \to Z(A)$. I would like to gain more intuition of what a $k$-algebra is, I've seen that one can give a ring $A$ a $k$-module structure by defining $\alpha.a=\rho(\alpha)a$.
My first question is: if I have a ring $A$ seen as left a $k$-module, where the action comes from the ring morphism $\rho$, then wouldn't $A$ be also a right module and wouldn't $A$ as a left module be equal to $A$ as a right module? I ask this because $k.a=\rho(k)a=a\rho(k)$, so I could define $a.k=a\rho(k)$ and then $a.k=k.a$.
My second question is: is there another motivation on why we are interested in $k$-algebras appart from the fact that if A is a $k$-algebra, then we can give $A$ a $k$-module structure?
The reason we are interested in algebras is that lots of the rings we are interested in are algebras!
About your first question: yes. In fact, the condition that the morphism has image in the center is precisely that the left and right module structures coincide, and is in fact phrased in thay often.