Of course I did not look at their original work... :( Let's see.

Exceptionally, I don't think the downvote is right here...

Wow, the second one would be very creative.

Okay, you said "non-science majors". Probably Bourbaki is mathematically too advanced and historically not detailed enough. I'm sorry for this superfluous comment. :)

What about the historical appendices in Bourbaki? (This is more a question than a hint).

I think one source of this problem are definitions (in the first lectures) like: a bijective morphism of groups is called an isomorphism. Introducing categories (very roughly!), defining the general notion of an isomorphism in a category and mentioning that it's awesome that for groups we just have to check bijectivity could really prevent this...

@Hanno: Wow, entering "katenär" in google gives 16 hits (already including your answer :)). I have never heard/seen this word; but on the one hand I never talked about these rings in German and on the other hand google books gives at least 3 relevant hits, so it seems that some mathematicians indeed use it!

Oh, it's Rasmus' link, sorry.

@captain obvious: Well, the one who defined what an algebra over a commutative ring is decided this! Take a look at your wikipedia link: ...is a morphism $f:R \rightarrow A$ such that the image of $f$ is contained in the centre of $A$. But I think this makes sense.

But then I would call $A$ an "$R$-Ring" (I'm thinking of the comma category Rng/R) rather than an "$R$-algebra". But, again, I have no idea...

The problem with that definition is that it is not a generalization of an algebra over a commutative ring because here R is mapped to Z(A).

I just asked the same. Does this definition imply that $R$ is central in $A$? If so, then I think it's not the right definition. I think a ring morphism $R \rightarrow A$ could be a "correct" definition, but I also have no idea.

I remember my question, but there it was about algebras over commutative rings. That's fine!