Notes:Modules

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Motivation

Let (V,F) be a vector space over the field F. Let τL(V)[Note 1] be given. Then for any polynomial, p(x)F[x] (where F[x] denotes the space of polynomials over the field F) the operator:

  • p(τ) is well defined.
    • For example, consider p(x)=1+2x+x3 then p(τ)=i+2τ+τ3 where i:VV is the identity map, and τ3 is the threefold function composition τττ

Thus using the linear operator τ:VV we can define the product of a polynomial, p(x)F[x] and a vector, vV by:

  • p(x)v:=p(τ)(v) meaning p(x)v:=(p(τ))(v) of course.

Clearly this "product" satisfies the usual properties of scalar multiplication, namely:

  • For r(s),s(x)F[x] and u,vV:
    1. r(x)(u+v)=r(x)u+r(x)v
    2. (r(x)+s(x))u=r(x)u+s(x)u
    3. [r(x)s(x)]u=r(x)[s(x)u] and
    4. 1u=u

Thus for a fixed τL(V) we can think of V as being imbued with operations of addition and multiplication of an element of V with an element of F[x]!

Of course, F[x] is not a field so these operations cannot be the operations of a vector space and so cannot be an alternate vector space structure on V.

This leads us to modules:

Definition

Let R be a cu-ring[Note 2], then[1]:

  • We may call the elements of R scalars (but not of a vector space, of a module instead!)

An "R-module" or "module over R" is a non-empty set M, together with two operations[1]:

  1. Addition: +:M×MM by +:(u,v)u+v as discussed, and
  2. Multiplication[Note 3]: ×:R×MM by ×:(r,v)rv as discussed.

We call R the base ring of M[1]

Immediate properties

Immediately we see the following[1]:

  1. M is an Abelian group under addition
  2. For all r,sR and u,vM the following hold:
    1. r(u+v)=ru+rv
    2. (r+s)u=ru+su
    3. (rs)u=r(su)
    4. 1u=u

Examples


TODO: Check these for requirements of the ring. See "importance of the base ring" below


  1. If R is a ring then the set Rn of all ordered n-tuples of elements of R is an R-module, with addition and multiplication defined component wise (just as for Fn - the n-dimensional vector space of a field)
    • Caution:I think the terminology has changed here, it might mean a ring with unity... I'll check later!
    • For example Zn is the Z-module of all ordered n-tuples of integers.
  2. If R is a ring then the set Mm,n(R) of m×n matrices with elements of R is an R-module, with the usual operations.
    • Since R is a ring we can also take the product of matrices in Mm,n(R)
    • We can also take R=F[x], where Mm,n(F[x]) is the F[x]-module of all m×n matrices whose entries are polynomials
      • Caution:...What... check all of this for any implicit requirements on the rings
  3. Any cu-ring is a module over itself. That is R is an R-module. Scalar multiplication is just the ring multiplication.

Importance of the base ring

This definition requires that R be commutative.

Modules of non-commutative rings are a "thing".

I must be very careful when "distilling" this definition

Notes

  1. Jump up Here we consider L(V,W) as the set of all linear transforms from V to W. Not just the continuous ones. This distinction doesn't matter if the vector spaces are finite (or if just V is finite Caution:I would have thought)
  2. Jump up Recall this means:
  3. Jump up Which we denote by "juxtaposition", so no operator, eg ab rather than a×b or ab or ab or whatever

References

  1. Jump up to: 1.0 1.1 1.2 1.3 Advanced Linear Algebra - Steven Roman