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, Q is a Z -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
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 AB (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 Z (a free Z-module) is called a free abelian group.
Free module | Free abelian group | Projective module | Flat module | Galois module
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Projective module
Kaplansky's theorem
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Irving Kaplansky (1917-2006)
Flat module
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Galois module
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Flat module (1956) | Jean-Pierre Serre (1926-)
