Search results
Results from the WOW.Com Content Network
For any non-negative integer n, =, the cartesian product of n copies of R as a left R-module, is free. If R has invariant basis number, then its rank is n. A direct sum of free modules is free, while an infinite cartesian product of free modules is generally not free (cf. the Baer–Specker group).
Free A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces. Projective Projective modules are direct summands of free modules and share many of their desirable properties. Injective
A free module is a module that can be represented as a direct sum over its base ring, so free abelian groups and free -modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free -module, and each free -module comes from a free abelian group in this way. [21]
The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module as a direct sum of submodules.
The tensor product of two modules A and B over a commutative ring R is defined in exactly the same way as the tensor product of vector spaces over a field: := /, where now () is the free R-module generated by the cartesian product and G is the R-module generated by these relations.
A 21-year-old Norwegian woman has become the youngest person to reach the South Pole on skis, solo and without assistance, her team told AFP on Tuesday. Karen Kylleso accomplished the feat ...
A module that has a basis is called a free module. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions. A module over the integers is exactly the same thing as an abelian group. Thus a free module over the integers is also a free abelian group.
Karyn Hascal, The Healing Place’s president and CEO, said she would never allow Suboxone in her treatment program because her 12-step curriculum is “a drug-free model. There’s kind of a conflict between drug-free and Suboxone.” For policymakers, denying addicts the best scientifically proven treatment carries no political cost.