R-Module Structure In Chapter 10 of Dummit & Foote (3rd Edition), the authors make remarks about the "R-module structure" of things. For instance, when talking about an R-Algebra 'A', he says:
"If A is an R-algebra then it is easy to check that A has a natural left and right (unital) R-module structure defined by r.a = a.r = f(r)a where f(r)a is just the multiplication in the ring A. In general it is possible for an R-algebra to have other left or right R-module structures, but unless otherwise stated, this natural module structure on an algebra will be assumed."
I'm just a little confused about this statement, in particular the part about the 'natural left and right R-module structure' is what confuses me. Just looking for some clarification. Thanks!
 A: It all depends on your definitions : for some authors, a $R$-algebra $A$ is a $R$-module which is also a ring, such that multiplication is $R$-bilinear. For others, it is a ring $A$ with a ring morphism $R\to Z(A)$ ($Z(A)$ being the center of $A$). You can also come up with other (equivalent) definitions.
If you start from the second definition I gave, then there is no module structure mentioned. So you have to check that $A$ is indeed a (left and right) $R$-module.
Think of the $\mathbb{R}$-algebra $\mathbb{C}$. You can see it a a ring $\mathbb{C}$ with a morphism $j: \mathbb{R}\to \mathbb{C}$. But then it is also a $\mathbb{R}$-vector space, and the vector space structure is given by multiplication inside $\mathbb{C}$. The same thing happens for any algebra over any commutative ring.
You may also be confused by the fact that the author checks both left and right module conditions, since when the base ring is commutative, one usually only considers left modules, because left and right modules are basically the same thing over a commutative ring. But it can happen that a given abelian group $M$ has a different structure of left and right module over a commutative ring $R$. Here the module structure comes from multiplication inside the agebra. But the elements of $R$ can act by multiplication either on the left or on the right. Thankfully, this is the same thing since the morphism $R\to A$ has image in the center.
