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I had previously seen the definition to be the one in Atiyah-Macdonald's Commutative Algebra:

  1. 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?

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    @RickyDemer done2012-12-07

1 Answers 1

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The difference between the two definitions of algebra over $A$ is the Atiyah-Macdonald wants all rings to have a unit, all ring homomorphisms to be unitary and all algebras to be associative. The definition from wikipedia does not imply this. If we add this to the definition (2), that is rephrase it as

An $A$-algebra $B$ is an $A$-module $B$ together with an $A$-bilinear associative $[,]\colon B^2\to B$ such that $[,]$ has a unit.

we get the desired $\phi$ by letting $\phi(a) = a1_B$ where the multiplication denotes the $A$-module operation on $B$.