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Tag Archives: homomorphism
Elementary Module Theory (III): Approaching Linear Algebra
The Hom Group Continuing from the previous installation, here’s another way of writing the universal properties for direct sums and products. Let Hom(M, N) be the set of all module homomorphisms M → N; then: (*) for any Rmodule N. In the case where there’re finitely … Continue reading
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Tagged cokernels, direct products, direct sums, homomorphism, isomorphism theorems, kernels, linear algebra, modules, submodules, vector space
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Elementary Module Theory (II)
Having defined submodules, let’s proceed to quotient modules. Unlike the case of groups and rings, any submodule can give a quotient module without any additional condition imposed. Definition. Let N be a submodule of M. By definition, it’s an additive … Continue reading
Posted in Notes
Tagged cokernels, direct products, direct sums, homomorphism, isomorphism theorems, kernels, modules, submodules
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Introduction to Ring Theory (4)
It’s now time to talk about homomorphisms. Definition. Let R, S be rings. A function f : R → S is a ring homomorphism if it satisfies the following: f(1R) = 1S; f(x+y) = f(x) + f(y) for all x, y in … Continue reading
Casual Introduction to Group Theory (6)
Homomorphisms [ This post roughly corresponds to Chapter VI of the old blog. ] For sets, one considers functions f : S → T between them. For groups, one would like to consider only actions which respect the group operation. Definition. Let G and … Continue reading
Posted in Notes
Tagged advanced, factor through, group theory, homomorphism, isomorphism theorems, normal subgroups, universal properties
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