In mathematics, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module,[1] but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.
Given any set S and ring R, there is a free R-module with basis S, which is called the free module onS or module of formalR-linear combinations of the elements of S.
A free abelian group is precisely a free module over the ring Z of integers.
^Keown (1975). An Introduction to Group Representation Theory. p. 24.
projective modules enlarges the class of freemodules (that is, modules with basis vectors) over a ring, keeping some of the main properties of freemodules. Various...
stably freemodule is a module which is close to being free. A module M over a ring R is stably free if there exists a free finitely generated module F over...
generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over...
Look up module or modular in Wiktionary, the free dictionary. Module, modular and modularity may refer to the concept of modularity. They may also refer...
field or a freemodule over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions...
algebra, flat modules include freemodules, projective modules, and, over a principal ideal domain, torsion freemodules. Formally, a module M over a ring...
Then M/tM is a finitely generated torsion freemodule, and such a module over a commutative PID is a freemodule of finite rank, so it is isomorphic to R...
equivalently be called free Z {\displaystyle \mathbb {Z} } -modules, the freemodules over the integers. Lattice theory studies free abelian subgroups of...
algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring...
finitely generated free K[x] module is also free. The construction of higher order syzygy modules is generalized as the definition of free resolutions, which...
abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
a free commutative algebra. For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X1,...,Xn} is the free R-module with...
e.g. M itself, this shows that a module is a quotient of a freemodule, a useful fact.) A generating set of a module is said to be minimal if no proper...
left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For...
A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological...
ring is a freemodule and at the same time a ring, constructed in a natural way from any given ring and any given group. As a freemodule, its ring of...
loadable module (kld) in FreeBSD, kernel extension (kext) in macOS (although support for third-party modules is being dropped), kernel extension module in AIX...
modules is torsionless. A freemodule is reflexive if it is finitely generated, and for some rings there are also infinitely generated freemodules that...
The Apollo command and service module (CSM) was one of two principal components of the United States Apollo spacecraft, used for the Apollo program, which...
of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction...
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