Difference between algebra and module?

[precondition]

Obviously having been assigned two different terms indicates these two objects are different but as far as I know,
algebra: vector space with the multiplication which takes two elements in vector space and give another in vector space
module: vector space over ring(extending from field)

hmm... i guess what i want to know is "is algebra an example of module?"
in addition could you sort of give me an example or otherwise so that I can concretely distinguish the two?

I appreciate your help!

 

[Hurkyl]

Algebra is a term used in very general situations.

But in the more common usage you describe, yes, every algebra over a ring is a module over that ring.


As for the vector space / module thing, the correct fact is

a module over a field is a vector space;

"module" is the general term; "vector space" is the special case.

 

[matt grime]

A module is something that an algebra (or ring) acts on as endomorphisms. An algebra is a module, since the algebra acts on itself by mutliplication.

Modules, in general, need not be defined over a field, vector spaces usually are required to be over a field.

A module for the ring Z is just an abelian group, for example. A module for a field, F, is the same as a vector space over that field. But for more complicated objects, whilst a module for an algebra is a vector space, not every vector space is a module, and part of the definition is the action of the algebraic gadget. So a module is not just a vector space, it is one with extra structure.

 

[DeadWolfe]

Hmm. Not every vector space is a module? What definition of module do you go by?

 

[HallsofIvy]

Obviously having been assigned two different terms indicates these two objects are different but as far as I know,
algebra: vector space with the multiplication which takes two elements in vector space and give another in vector space
module: vector space over ring(extending from field)

That pretty much says it all. In an "algebra" we have a vector space with a multiplication of vectors defined. In a "module" there is not necessarily any such multiplication defined. I don't see why you would think that being "over a ring" rather than "over a field" would suddenly create a multiplication operation!

hmm... i guess what i want to know is "is algebra an example of module?"

No, an algebra is still defined over a field, not a ring.

in addition could you sort of give me an example or otherwise so that I can concretely distinguish the two?

I appreciate your help!

The set of all polynomials over the real numbers is an algebra that is not a module.

The set of all pairs (x,y) with addition defined by (x,y)+ (u,v)= (x+u,y+v), x, y, u, v all members of Z₄ (integers mod 4) is a module that is not an algebra.

 

[matt grime]

Hmm. Not every vector space is a module? What definition of module do you go by?

I realize I phrased this poorly - I even said that every vector space over F is a module for F. Linear algebra is just the representation theory of a field. Let me try to say it properly: a module is a module for something. A vector space alone is not a module - you need to say what it is a module for, or over. For instance C is a rank two module over R, or rank one over C: you need to say more than 'is a vector space is a module'. A vector space is a module in more than one way - infinitely many in fact. Uncountably infinitely many. It is a module for a proper class of non-isomorphic objects in fact.

 

[mathwonk]

multiplication.

tangent vector fields are a module over the ring of functions on a manifold, while the exterior forms are an algebra.

if R is a commutative ring, and S is any ring, a ring map R-->S with image in the center of S, makes S an R algebra.

If G is an abelian group, then S = End(G) is a ring, and an R algebra structure R-->S in this case, makes G an R module.

I.e. an R module struture on G, is an R algebra structure on End(G).

is this right matt? (I am not the specialist here.)