Regular module endomorphisms into itself

Question

Let $k$ be a field and let $A$ be an algebra over $k$. Denote by $End_A (A)$ the set of all endomorphisms of the regular $A$-module $A$ into itself. Fix $a \in A$, and define the A-module homomorphism $r_a : A \rightarrow A$ by $r_a(x) = x \cdot a$.

Clearly, $\{r_a: a \in A\} \subseteq End_A (A)$. But, is it true that moreover $\{r_a: a \in A\} = End_A (A)$? If so, how to show this?

Answer 1

Hint: An $A$-module endomorphism is determined by where it sends $1\in A$.