associative R-algebra A, center

Question

I would like to know why here http://en.wikipedia.org/wiki/Algebra_%28ring_theory%29

in the sentence

An equivalent definition of an associative R-algebra is a ring homomorphism f:R\to A such that the image of f is contained in the center of A.

im f should be contained in the center of A. Which property of an associative R-algebra A this requires. JG

Answer 1

This wiki article is poor, I recommend looking for an alternative source.

Suppose we have a ring map $f: R \to A$. For me, rings always have unit elements and ring homomorphisms must send the unit of one ring to the unit of the other. This means that $A$ has a unit element. We make $A$ into a left $R$-module by $r \cdot a := f(r)a$ for $r \in R, a \in A$. I claim that if this makes $A$ into an $R$-algebra according to the definition from the section "formal definition", then the image of $f$ lies in the centre of $A$.

The second bilinearity condition from the "formal definition" section is saying that $$ a( r\cdot b + s \cdot c) = r \cdot (ab) + s \cdot (ac) $$ for all $a,b,c \in A, r,s \in R$. Taking $c=0, b=1$ we have $a(r\cdot 1)= r\cdot a$, that is $af(r) = f(r)a$, thus $f(r)$ is in the centre.