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 \(\mathbb Z\)-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.

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