Equivalent definitions of algebra over a ring

Question

I'm trying to see how the following definitions of an algebra A over a ring R are equivalent. We have:

1) An algebra is a ring A which is also an R-Module such that the ring multiplication and module multiplication are compatible. This means that for $x,y \in A$ and $r \in R$ we have $r(xy)=(rx)y=x(ry)$

2) An algebra is a ring homomorphism $\phi: R \to Z(A)$

I see how an homomorphism in (2) gives rise to an algebra in definition (1) under the module multiplication of $r \cdot x=\phi(r)x$. However, I don't see how an algebra from (1) gives a homomorphism in (2). Why must the module multiplication by an element $r$ behave like ring multiplication for some element of A?

Answer 1

To go from 1) to 2) you can define $\phi$ by $\phi(r)= r1$.