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A free module is a module with a basis. [2] ... while an infinite cartesian product of free modules is generally not free (cf. the Baer–Specker group).
A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r = 0 or m = 0. Noetherian A Noetherian module is a module that satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes ...
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]
Thus the Cartesian product G × H is equipped with the structure of an abelian group by defining the operations componentwise: (g 1, h 1) + (g 2, h 2) = (g 1 + g 2, h 1 + h 2) for g 1, g 2 in G, and h 1, h 2 in H. Integral multiples are similarly defined componentwise by n(g, h) = (ng, nh) for g in G, h in H, and n an integer. This parallels ...
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.
KNOXVILLE, Tenn. (AP) — Zee Spearman scored 18 points, including a basket with 12 seconds left, to lead No. 19 Tennessee to an 80-76 win over No. 5 UConn on Thursday night. The Lady Volunteers ...
Witnesses have revealed what they saw take place at the moment of impact during the horrifying collision between an American Airlines jet and an Army helicopter over the Potomac River. On ...
For example, modules need not have bases, as the Z-module (that is, abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field , with the elements being called vectors.