Abstract Nonsense

Homomorphisms Between Finitely Generated Abelian Groups (Pt. I)

Point of Post: In this post we derive the necessary information to describe $\text{Hom}_\mathbb{Z}(A,B)$ for any finitely generated abelian groups $A,B$ according to their cyclic group decomposition.

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Motivation

In this post, using our ability to split Hom across finite products we shall show how to effectively calculate the Hom group between any two finitely generated abelian groups. The only caveat to this is that one must first decompose the finitely generated abelian groups into their cyclic decomposition as is guaranteed by the structure theorem. Probably the most useful aspect of this post is that it will, if you keep your bookkeeping orderly, the number of homomorphisms between two finite abelian groups (presented as a product of cyclic groups) and the generators that will enable one to, in theory, produce every single homomorphism.

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November 14, 2011

Triangle Completions

Point of Post: In this post we discuss when certain mapping triangles of modules may be completed.

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Motivation

The concept I’d like to discuss in this post is the more theoretical way of viewing mappings, in regards to “factoring through” (i.e. completing a mapping triangle). Said differently, given mappings in and out of a given module, say $f:A\to B$ and $g:A\to C$ when can we find a morphism $h:C\to B$ which makes the resulting triangular diagram commute? We shall see that this will be of great importance to us when we start getting into some of the more advanced module theoretic topics. It mostly makes proofs easier by being able to, given a diagram, just decide whether or not we can complete it.

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November 2, 2011

Homomorphism Group (Module) of Two Modules (Pt. I)

Point of Post: In this post we discuss how given two left $R$-modules one can construct the set $\text{Hom}_R(M,N)$ of all $R$-morphisms $M\to N$ and how this is naturally an abelian group, and how under the right conditions (commutativity of $R$) it is also a left $R$-module. Also, we define the notion of an $R$-algebra (a $R$-module which is also a ring, with multiplication compatible with scalar multiplication) and show how under the same assumptions the set $\text{End}_R(M)$ is naturally a $R$-algebra.

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Motivation

We know from the study of vector spaces that an extremely important object associated to every $k$-vector space $V$ was the dual space $V^\ast$ of all linear functionals $V\to k$.  Of course, after a little notation one is able to renotate the dual space as the set $\text{Hom}(V,k)$ of all $k$-morphisms $V\to k$ when $k$ is considered as a one-dimensional vector space over itself. In fact, not long after this one realizes that given any other $k$-space $W$ one can form the set $\text{Hom}(V,W)$ of $k$-morphisms $V\to W$, which turns out to be a vector space of dimension $\dim_k(V)\dim_k(W)$ when $\dim_k(V),\dim_k(W)<\infty$. We saw that the studying of these spaces were not only useful for outward looking facts about linear algebra (e.g. direct applications of them, using them for their own sake) but also gave us structural information about $V$ as a vector space. Indeed, one can recall that $\text{Hom}(V,F)$ had the ability to detect the finite-dimensionality of $V$ since $V\cong \text{Hom}(V,F)$ if and only if $\dim_k(V)<\infty$. This shall be a recurring fact that we shall be discussing extensively in the math-to-come, to be particular, we will see that a general $R$-module $M$ can be studied quite extensively by watching how the “function” (functor) $\text{Hom}_R(M,\bullet)$ as $\bullet$ varies over other left $R$-modules. Consequently, we begin (just the very beginning) looking at $\text{Hom}_R(M,N)$ for two left $R$-modules. We show that it’s always an abelian group, under certain conditions a left $R$-module, and when insist that $M=N$ so that we get the set $\text{End}_R(M)$ of endomorphisms it’s (under the same previously mentioned conditions) an $R$-algebra (module which is also a ring). We also then discuss (practically just define) the “function” (functor) we previously mentioned and show some of its most basic properties (i.e. that it is a functor).

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November 1, 2011

Ring Homomorphisms (Pt. II)

Point of Post: This post is a continuation of this one.

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June 18, 2011

Ring Homomorphisms (Pt. I)

Point of Post: In this post we discuss the notion of ring  homomorphisms, kernels, images, etc.

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Motivation As is standard in math, especially in algebra after defining a structure and its subobjects we define the morphisms between the two objects. In particular, we would now like to define the morphisms between two rings $R$ and $R'$. What do we want these morphisms to do? Well, we clearly want the morphisms to preserve the ring structure. In other words, we’d like it to be additive (be a group homomorphism for the group structure) and multiplicative (be a semigroup homomorphism for the semigroup structure). After this we define the obvious notions of kernel and image and show that they are, in fact, subrings ,etc.

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June 18, 2011

Interesting Theorem Regarding Linear Transformations

Point of post: In this post we discuss an interesting result which tells us precisely when a mapping from one vector space to another is a linear transformation, namely if and only if the graph (to be defined below) is a subspace of the direct sum of the two vector spaces.

Motivation

As someone who has done the majority of their mathematical work in topology I can say I am well acquainted with the innocuous concept of the graph of a mapping playing an important role in the theory of structure preserving maps. There is the Closed Graph Theorem in functional analysis, the fact that a function from $f:X\to Y$ where $X$ is Hausdorff and $Y$ is compact is continuous if and only if the graph is closed in $X\times Y$ with the product topology ,etc. That said, I had no idea, until now, that there is a simple but satisfying analogue for linear transformations. Namely, if $\mathscr{V}$ and $\mathscr{W}$ are $F$-spaces and $T:\mathscr{V}\to\mathscr{W}$ we may define the graph, denotes $\Gamma_T$, to be the set $\left\{\left(v,T(v)\right):v\in\mathscr{V}\right\}$. Then, $T\in\text{Hom}\left(\mathscr{V},\mathscr{W}\right)$ if and only if $\Gamma_T$ is a subspace of $\mathscr{V}\boxplus\mathscr{W}$.

So, that being said, all I have left to say is the proof of this interesting theorem:

November 22, 2010