Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    17
    ReputationRep:
    '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.
    Offline

    2
    ReputationRep:
    (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 R is a ring then a module is an abelian group upon which R acts.

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

    For instance, if the ring R is a field (think of \mathbb{R} or \mathbb{C} 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 R is the ring R 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 R. 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 D_n. 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 D_n 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 GL_2(\mathbb{R}) of invertible 2x2 matrices acting on the plane \mathbb{R}^2 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.
    Offline

    15
    ReputationRep:
    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.
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Brussels sprouts
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.