Final Answers
© 2000-2020 Gérard P. Michon, Ph.D.

Modules

Modules are vectorial structures over a ring of scalars (instead of a field).
Free modules have a basis similar to that of vector spaces.
Injective modules. The rationals form an injective module over the integers.
Projective modules. Due to Eilenberg & Cartan (1956).
Flat modules. Devised by Jean-Pierre Serre in 1956.

Module Theory

(2006-03-28) Module over a Ring A A vectorial structure where division by a scalar isn't "well defined".

A module obeys the same basic rules as a vector space, but its scalars are only required to form a ring; a nonzero scalar need not have a reciprocal...

A module over A may be called an A-module. For example, is a -module. This is to say that the rationals form a module over the integers (this particular example gave birth to the concept of an "injective module").

Module | Free module | Projective module | Flat module | Galois module

(2020-05-24) Free Modules They have a vector-like basis. Not all modules do.

Given a ring A and a set B, the free A-module of basis B consists of all formal linear combinations of elements of S, with coefficients in A.

It's understood that a formal linear combination is zero only when all its coefficients are. So the above definition means that all elements of B are linearly independent in the module so generated. Just like in the case of vector spaces, what we call a basis is a linearly-independent set of generators. Recall that a linear combination is only allowed to have a finite number of nonzero coefficients. The above free module is denoted:

A^(B)

If B is infinite, that's much smaller than the module A^B (consisting of all applications from B to A, which can be added and scaled pointwise) which B is much too small to generate!

A free module over (a free -module) is called a free abelian group.

Free module | Free abelian group | Projective module | Flat module | Galois module

(2020-05-26) Injective Modules The rationals form an injective module over the integers.