In Dummit and Foote, they define an R-Algebra Homomorphism as a ring homomorphism with the added condition that $\phi(r\cdot a)=r\cdot \phi(a)$. But in the previous part Dummit mentioned that there could be multiple R-module structures for an R-Algebra.
My question is, is the $\cdot$ product defined from the natural module structure? Or is the definition more general in the sense that I choose the Module structure first before defining the R-Algebra Homomorphism.
Please tell me if I did not express my question in a clear manner.
Thanks in advance for any feedback.
Best Answer
You can either define a $R$-algebra $A$ to be a ring with $R$-module structure, such that the multiplication is an $R$-linear if you fix a component. Or in case that $A$ has $1$ you can require $A \otimes_R A \to A$ to be $R$-linear, since then $A$ has a natural $R$-bimodule structure.
I think this definition has some disadvantages. First it depends on the definition of ring. Is it associative, commutative, has it a unit? Rings often are assumed to have a unit and are mostly associative. In case of an algebra you don't want this always.
An other point is that in most cases you have modules which you equip with a multiplication. For example in the case of polynomials, they have a quite natural module structure and you then define a multiplication.
So I prefer the definition a $R$-algebra is a $R$-module $A$ equipped with a $R$-bilinear map $A\times A \to A$. And then you can require this map to be associative, unital, commutative or whatever you need.
Now to answer you question. Mostly I guess you have a modules and equip it with an algebra structure. Then you usually want the natural structure of the module. And in the definition of algebra is the module structure included, so if you have an algebra that data is fixed.
You could define an other algebra from that by defining an other scalar multiplication, but this then is an other algebra.
Of cause you can also take a ring and then define the module structure, but then you should't think about the natural module structure, which this algebra may also has. Just the one you choose wich is included in the data of the algebra.