Are All Modules Merely Vectors, or Do They Vary by Type and Operation?

[pivoxa15]

I know there are many examples of rings like R[x], N, Q, where the elements can be fundalmentally different like polynomials and numbers.

But are there different type of modules like there are rings? Or are modules just any vector? And the modules are different when you consider which R module they are. So N-Module is different to N[x]-Module. But all N-Modules are identical?

 

[Hurkyl]

I know there are many examples of rings like R[x], N, Q, where the elements can be fundalmentally different like polynomials and numbers.

But are there different type of modules like there are rings? Or are modules just any vector? And the modules are different when you consider which R module they are. So N-Module is different to N[x]-Module. But all N-Modules are identical?

Every abelian group is a Z-module, for example.

Every N[x]-module can also be viewed as an N-module.

Edit: I'm assuming either:
(1) N was just a variable denoting a ring
(2) N is the natural numbers and you're talking about modules over semirings. (N, of course, isn't a ring)

 

[pivoxa15]

I see what you mean but I was referring to what kind of things are the modules themselves without considering the scalars in the ring R.

In the definition, you have
A left R-module over the ring R consists of an abelian group (M, +) and an operation R × M → M (called scalar multiplication)

So the module without considering the scalar is any abelian group (M, +). So the module could be (Z, +) or (matrix, +) or (R+R, +) where + between the R's denote direct sum etc.

 

[Hurkyl]

I see what you mean but I was referring to what kind of things are the modules themselves without considering the scalars in the ring R.

In the definition, you have
A left R-module over the ring R consists of an abelian group (M, +) and an operation R × M → M (called scalar multiplication)

So the module without considering the scalar is any abelian group (M, +). So the module could be (Z, +) or (matrix, +) or (R+R, +) where + between the R's denote direct sum etc.

If a module is an abelian group with extra structure, and you discard the extra structure, then you simply have an abelian group. I guess I don't understand the point of your question.

 

[pivoxa15]

Good point. Modules are nothing more than abelian groups with ring scalars that interact via a few axioms.

 

[pivoxa15]

There is no such thing as multiplying two modules together is there since modules are additive abelian groups only. So they can only add with each other.

 

[Hurkyl]

There is no such thing as multiplying two modules together is there since modules are additive abelian groups only. So they can only add with each other.

You mean that a generic module doesn't have a multiplication operation for its elements.

There are at least two useful ways to multiply modules: the Cartesian product and the tensor product.