I am trying to connect the definition of an algebra from baby rudin to a chapter in an abstract algebra text. It seems however, that an R-algebra isn’t the same as what rudin is talking about. What is the more precise name for the algebra given my Rudin?
Definition (Dummit and Foote) Let $R$ be a commutative ring with identity. An $R$ -algebra is a ring $A$ with identity together with a ring homomorphism $f: R \rightarrow A$ mapping $1_{R}$ to $1_{A}$ such that the subring $f(R)$ of $A$ is contained in the center of $A$.
7.28 Definition (Rudin PMA) A family $\mathscr{A}$ of complex functions defined on a set $E$ is said to be an algebra if (i) $f+g \in \mathscr{A}$, (ii) $f g \in \mathscr{A}$, and (iii) $c f \in \mathscr{A}$ for all $f \in \mathscr{A}, g \in \mathscr{A}$ and for all complex constants $c$, that is, if $\mathscr{A}$ is closed under addition, multiplication, and scalar multiplication. We shall also have to consider algebras of reai functions; in this case, (iii) is of course only required to hold for all real $c$. If $\mathscr{A}$ has the property that $f \in \mathscr{A}$ whenever $f_{n} \in \mathscr{A}(n=1,2,3, \ldots)$ and $f_{n} \rightarrow f$ uniformly on $E$, then $\mathscr{A}$ is said to be uniformly closed. Let $\mathscr{B}$ be the set of all functions which are limits of uniformly convergent sequences of members of $\mathscr{A}$. Then $\mathscr{B}$ is called the uniform closure of $\mathscr{A}$. (See Definition 7.14.)
Best Answer
For a possibly useful broader context:
What "$R$-algebra $A$" means is very context dependent, and there are many mutually incompatible conventions... so you just have to hope that either your source explains what they mean, or you can infer it from context.
In particular, it is essentially a waste of time to worry too much about comparison, much less reconciliation, of various versions.
For that matter, must a ring have a unit $1$? :) Certainly many attractive theorems use existence of $1$. But this can be weakened to "existence of sufficiently-many idempotents", meaning that for any finite (for example) subset $X$ of the ring, there is an idempotent $e$ (meaning that $e^2=e$, such that $ex=x$ for all $x\in X$. If the ring is not commutative, then we may want to specify left/right conditions.
Similarly, for a commutative ring $R$ with $1_R$, for a ring $A$ to be an "$R$-algebra", do we really need $R$ to inject to $A$? After all, $\mathbb Z/n$ is a pretty reasonable $\mathbb Z$-algebra.
Do we really need the image of $1_R$ in $A$ to be $1_A$? Or merely that $1_R\cdot a=a$ for all $a\in A$?
And so on.
I've come to think that there's no universally optimal definition of "algebra", but, rather, that there are many somewhat-different things that can be called "algebras", and needing some explanation for clarity. So it's not so much "definitions", but just "naming".