We can define an $A$-algebra $R$ as a ring R with a map $A \to Z(R)$. Equivalently, we can be slightly more restrictive and only consider fields where we have a ring $R$ and a field $F$ and $R$ is an $F$-algebra if $$c(xy)=(cx)y=x(cy)$$
Question:
Is it not true that, say
$$x(cy)=(xc)y$$
since $c$ can be thought of as (or identified with if you prefer) an element of $Z(R)$? I know we have left/right modules but there doesn't (to me) seem to be an equivalent identifications for algebras. I do know that $R$ can be thought as an $F$-module. What I am thinking then is that we could equivalently write the multiplication on the right and we have made a notational choice only. Since our ring of scalars will always be commutative any left module is also a right module. Is this the idea or am I missing something?
Related Post(s):
Modules over an algebra: confusion in understanding "left" and "right" business