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# Modules, commutative algebra Watch

1. 'A left R-module M over the ring R consists of an abelian group (M, +) and an operation R × M → M (defined (r,m)-->rm) such that for all r, s in R and x, y in M, we have:....'

Ok, I sort of get what a module is. The thing that bugs me, and I cannot get past, is the: operation R × M → M (defined (r,m)-->rm).

What I currently understand, is that a module is an abelian group with the + operation, and it follows certain properties (distributivity, associativity etc..)

Can someone shed some light on the operation bit? I really don understand what its getting at. I have searched the internet, but they give no explanation.
2. (Original post by 2710)
'A left R-module M over the ring R consists of an abelian group (M, +) and an operation R × M → M (defined (r,m)-->rm) such that for all r, s in R and x, y in M, we have:....'

Ok, I sort of get what a module is. The thing that bugs me, and I cannot get past, is the: operation R × M → M (defined (r,m)-->rm).

What I currently understand, is that a module is an abelian group with the + operation, and it follows certain properties (distributivity, associativity etc..)

Can someone shed some light on the operation bit? I really don understand what its getting at. I have searched the internet, but they give no explanation.
Do you know what a group action is?

A module is basically a ring action. That is, if is a ring then a module is an abelian group upon which acts.

The only way to understand or get what these things are about is to look at examples.

For instance, if the ring is a field (think of or if you like) then a module is just a vector space over that field. This is a good way to initially understand the definition of a module but in general, it isn't a good example to think of in general since it is way too specialised.

The most obvious example of a module for a ring is the ring itself. Any ring is an abelian group and you can consider it acting on itself by multiplication. This module has a special name - it is called the regular representation of . You can also check that any ideal of a ring is a module where again, the ring acts by multiplication. This example is a good way to think about modules since in some sense, the idea of a module is a generalisation of an ideal in a ring.

The general idea is that a module for a ring is a representation of that ring. Any algebraic structure is somewhat like a collection of symmetries (I'll come back to this in a minute) and a representation of an algebraic structure is like viewing those symmetries as actually acting on something rather than just being relations between symbols.

Let me talk about group actions to explain what I mean. We are all familar with the dihedral groups . Now, we are normally introduced to this group in two ways, as a set of elements with some relations and also as the symmetries of the regular n-gon. The first is in essence the algebraic structure - the second is the action of that algebraic structure in some space. i.e. there is a group action of on the n-gon itself. By Cayley's theorem, every group is isomorphic to a subgroup of a symmetric group thus every group, even though it is a purely algebraic structure really is a set of symmetries of something.

Now, there is a straightforward analogy of Cayley's theorem for rings - every ring is isomorphic to a subring of the ring of endomorphisms of some abelian group. Again, this means that a ring is a set of symmetries that really act on objects external to them.

Along those lines, a great example to think about is a ring of matrices acting on a vector space for example think about the ring of invertible 2x2 matrices acting on the plane here the operation is simply the action of a matrix on a vector. Of course, this is an example of a module for a noncommutative ring but at least hopefully it makes you think about what the 'operation' of a module is all about.
3. I think it's easiest to view modules as akin to vector spaces, but with scalars now taken from a ring rather than a field. So the map (r,m) to rm is just scalar multiplication.

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Updated: March 29, 2013
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