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In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. [1]
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 on S or module of formal R-linear combinations of the elements of S.
Given an ideal I in a commutative ring R and an R-module M, the direct sum = / + is a graded module over the associated graded ring / +. A morphism f : N → M {\displaystyle f:N\to M} of graded modules, called a graded morphism or graded homomorphism , is a homomorphism of the underlying modules that respects grading; i.e., f ( N i ) ⊆ M ...
In 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 R , then a function f : M → N {\displaystyle f:M\to N} is called an R - module homomorphism or an R - linear map if for any x , y in M and r in R ,
As companies report every quarter, if you receive a statement from July 1 to Sept. 30, this would indicate how the company performed financially in the third quarter. Q4: The last quarter of the ...
In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1.
According to a study, our stress can rub off on our four-legged friends
Randy Moss, Cris Carter and Jake Reed formed one of the more famous receiver trios in NFL history. "Three Deep" put up a lot of yards and highlights for the Minnesota Vikings.