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 {M_k}, 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 {m_k} such that at most finitely many m_ks 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.

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