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(2006-03-28)  
Avectorial structure where division  by a scalar isn't "well defined".

A module  obeys the same basic rules as avector space,but its scalars  are only required toform aring; a nonzero scalar need not have areciprocal...

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").

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

Given a ring A  and a set B,  the free A-moduleof basis B  consists ofall  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 coefficientsare.  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 setof 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.

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

(2020-05-24)   (Eilenberg &Cartan, 1956)
A generalization of Free Modules.

(2020-05-24)     (Jean-Pierre Serre, 1956)