Module
over a Ring| Module
over a Ring |
In General
$ Left-module: An Abelian
group with a ring homomorphism f : R → Hom(X, X),
i.e., an
Abelian group X with a scalar multiplication R × X → X,
satisfying:
- 1. Distributivity in both factors: a (x+y) = ax + ay,
(a+b) x = ax + bx;
- 2. "Associativity":
(ab) x = a (bx).
$ Right-module: Can be defined directly by changing the appropriate
things in the definition, or as a left-module over the opposite of R; The
notions
of left- and right-module coincide if R is commutative.
* Remark: Property 1 by itself would make X into an R-group,
but R is
a ring, and we can require more structure.
* Generator: An element a 
A 
X such
that the smallest submodule of X containing A is X.
* Basis: A set of generators
such that every x in X is
a unique finite "linear" combination
of generators (linearly independent); Not every module possesses a basis, only
free ones.
Examples > s.a. types of modules.
- Every ring is a module over
itself;
- A Z-module is the same as abelian group;
- A Zk-module is an abelian group in which each element
has order a divisor of k;
- Vector or covector fields
on a manifold, over the ring of smooth functions;
- k-th order linear
differential operators between tensor densities of weight m and on
S1, over Diff(S1)
[@ Gargoubi et al JNMP(05)mp].
- If over a field, same as vector
space.
Direct Sum of R-Modules > s.a. category.
$ Def: Given a family
of R-modules {Mk},
where R is
a unitary ring (generalizable?), the direct sum 
k Mk is
the submodule of Xk Mk consisting
of those families {mk}
such that at most finitely many mk's
are non-zero; Operations are defined by (a, b)
+ (a', b'):= (a+a', b+b'),
(a, b):=
(a,
b).
* Examples: For a finite family of R-modules it is just the Cartesian
product; Vector spaces.
References > s.a. Module over
an Operad.
@ General: in Goldhaber & Ehrlich 70; in Hilton & Stammbach 71.
@ Torsion-free: Matlis 72.
@ Differential modules: Dubois-Violette m.QA/00-ln.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 jul 2008