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 \(R \times X \to X\),
satisfying:
- 1. Distributivity in both factors:
a (x+y) = ax + ay,
(a+b) x = ax + bx;
- 2. "Associativity":
(ab) x = a (bx).
$ Right-module: It 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 \(a \in A
\subset X\) 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 \(\mathbb Z\)-module is the same as abelian group.
- A \(\mathbb Z\)k-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 theory.
$ Def: Given a family of
R-modules {Mk},
where R is a unitary ring (generalizable?), the direct sum
\(\bigoplus_k M_k\) is the submodule of \(\times_k M_k\) consisting of
those families {mk} such
that at most finitely many mks
are non-zero; Operations are defined by (a, b)
+ (a', b'):= (a+a', b+b'),
λ(a, b):= (λa, λb).
* 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 97;
Keating 98.
@ Torsion-free: Matlis 72.
@ Differential modules: Dubois-Violette m.QA/00-ln.
> Online resources:
see Wikipedia page.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 13 feb 2016