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.)