I had previously seen the definition to be the one in Atiyah-Macdonald's Commutative Algebra:
- A is a ring and an algebra over a ring is a ring B such that there is a map $\phi:A\rightarrow B$.
which I tried to show is eqivalent to this definition(2) but I could not show that 2 implies 1. In particular, I think that the set of all polynomials over a ring with zero constant term satisfies 2 but not 1. How do I do this?