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
S^{1}, over Diff(S^{1})
[@ Gargoubi et al JNMP(05)mp].

- If over a field, same as vector space.

**Direct Sum of R-Modules** > s.a. category theory.

$

*

**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
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send feedback and suggestions to bombelli at olemiss.edu – modified 13 feb 2016