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Tag Archives: direct products
Commutative Algebra 25
Arbitrary Collection of Modules Finally, we consider the case where we have potentially infinitely many modules. Proposition 1. For a collection of A-modules , we have Proof First claim: we will show that the LHS satisfies the universal property for … Continue reading
Posted in Advanced Algebra
Tagged direct products, direct sums, exact sequences, local properties, local rings, localization
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Commutative Algebra 9
Direct Sums and Direct Products Recall that for a ring A, a sequence of A-modules gives the A-module where the operations are defined component-wise. In this article, we will generalize the construction to an infinite collection of modules. Throughout this article, let denote … Continue reading
Posted in Advanced Algebra
Tagged direct products, direct sums, modules, rings, universal properties
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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 R-module N. In the case where there’re finitely … Continue reading
Posted in Notes
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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Intermediate Group Theory (6)
In this post, we’ll only focus on additive abelian groups. By additive, we mean the underlying group operation is denoted by +. The identity and inverse of x are denoted by 0 and –x respectively. Similarly, 2x+3y refers to x+x+y+y+y. Etc … Continue reading
Posted in Notes
Tagged abelian groups, advanced, direct products, direct sums, free groups, generated groups, universal properties
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