TITLE: A module over an algebra. Is it a vector space?
QUESTION [18 upvotes]: Let $A$ be an algebra over a field $k$. I would like to know if my understanding of the following correct or not.
What I want to clarify is the definition of a module $M$ over $A$. I know the definition of a module over a ring. 

Is the definition of a module over an algebra $A$ the same as the
ring theoretic definition replacing a ring by an algebra?
Or, a module is a $k$-module, plus $A$ action?
Or, are they the same?

If it is the definition 2 above, then $M$ is a vector space over $k$. Is $M$ a vector space in the case of 1?

REPLY [16 votes]: Yes. A module $M$ over $A$ is just a module $M$ over the ring $A$; the additional structure of $A$ as a $k$-algebra plays no role.
(and 3.) That amounts to the same: since $A$ is a $k$-algebra, you already have a map $k \to A$ which turns $M$ into a $k$-module as well.

$M$, ultimately being a $k$-module as well, is a $k$-vector space. I'm not sure what you mean by "Is $M$ a vector space in the case of 1?"