Notes:Modules
Contents
[hide]Motivation
Let (V,F) be a vector space over the field F. Let τ∈L(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:V→V is the identity map, and τ3 is the threefold function composition τ∘τ∘τ
Thus using the linear operator τ:V→V we can define the product of a polynomial, p(x)∈F[x] and a vector, v∈V 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,v∈V:
- r(x)(u+v)=r(x)u+r(x)v
- (r(x)+s(x))u=r(x)u+s(x)u
- [r(x)s(x)]u=r(x)[s(x)u] and
- 1u=u
Thus for a fixed τ∈L(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]:
- Addition: +:M×M→M by +:(u,v)↦u+v as discussed, and
- Multiplication[Note 3]: ×:R×M→M by ×:(r,v)↦rv as discussed.
We call R the base ring of M[1]
Immediate properties
Immediately we see the following[1]:
- M is an Abelian group under addition
- For all r,s∈R and u,v∈M the following hold:
- r(u+v)=ru+rv
- (r+s)u=ru+su
- (rs)u=r(su)
- 1u=u
Examples
TODO: Check these for requirements of the ring. See "importance of the base ring" below
- 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.
- 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
- 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
- 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)
- Jump up ↑ Recall this means:
- R is a commutative ring with unity
- Jump up ↑ Which we denote by "juxtaposition", so no operator, eg ab rather than a×b or a∗b or a∘b or whatever