The role of maps in algebraic structures?

In group theory we have encountered the concepts of homomorphisms as maps from one group to another, and the same thing with ring and field theory and we ave called them respectively homomorphisms of groups (of rings, and of fields). Also in linear algebra we have linear maps. So my question is this: What is the role of such maps in this algebraic structures?

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Congratulations! You have just discovered category theory! –  MJD Jul 5 '12 at 16:29
Relevant: en.wikipedia.org/wiki/Morphism –  user2468 Jul 5 '12 at 16:31

One of my professors once said this about ring theory:

To study rings, you have two choices. You can take a particular ring, and stare at it intently for a long period of time until you discover interesting things about that ring. Then you write them down.

Or you can study how the rings "acts" on other structures, by studying the collection of all modules on that ring, and thus deduce interesting things about the ring. It turns out that the latter method is more fruitful, easier to generalize, and allows one to go from one ring to another ring more easily.

Studying the modules of a ring can be seen as a special case of studying homomorphisms (an $R$-module structure on an abelian group $A$ is equivalent to a ring homomorphism from $R$ to the endomorphisms of $A$, which are a ring under pointwise addition and composition. (Which is why I mention it)

The example of vector spaces is very apt. Vector spaces are nice objects, no doubt; but if all you do is stare at a particular vector space, you can say some interesting things (dimensions, subspaces, and so on), but it's not until you bring in linear transformations that the full power of vector spaces really becomes apparent. It is through linear transformations that we can actually do things with vector spaces: use them to solve systems of equations (linear and differential), to study Markov systems, to solve least squares problems (which amount to constructing certain kinds of projections, which are types of linear transformations), etc.

There is a whole philosophy of mathematics that says that the best way to study objects is to study the maps that "respect the structure", and not only for algebraic objects: topology is not just the study of topological spaces, it's the study of topological spaces and continuous maps between them. Analysis is not just the study of the real numbers, it is the study of the real numbers and real valued functions. Differential geometry concerns itself with differentiable maps. Etc. Turns out that maps give you a fruitful way of studying the structure.

In algebra, an important role is played by "congruences", which are equivalence relations on the underlying set that are compatible with the operations. For example, in the ring of integers, the usual modular congruences, $a\equiv b\pmod{n}$, give an equivalence relation that "respects" the ring structure on $\mathbb{Z}$, in the sense that if $a\equiv b\pmod{n}$ and $x\equiv y\pmod{n}$, then $a+x\equiv b+y\pmod{n}$ and $ax\equiv by\pmod{n}$ (the sum of classes is the same as the class of the sum; the product of classes is the same as the class of the product). Congruences correspnd, via the isomorphism theorems, to surjective morphisms.

Morphisms let you take one particular structure and study it from other points of view, in other contexts; it allows you (via "quotients") to focus on particular details while ignoring other part of the structure that may be "in the way" (e.g., the study of "parity" is made simpler by considering the quotient $\mathbb{Z}/2\mathbb{Z}$, rather than $\mathbb{Z}$ itself).

Morphisms let you understand inherent symmetry in an object; Klein proposed studying geometry almost exclusively in terms of certain kinds of symmetries and certain kinds of morphisms (projective transformations). Galois theory is founded on the idea of understanding the "symmetries" that may exist between different roots of the same polynomial (in the form of automorphisms).

All in all, morphisms are just a very rich way of studying objects. You can think of them as a way of letting you poke, examine, cut open, x-ray, fold, and generally manipulate your objects of interest so that you can get a better sense, not only of what they are, but how they interact with other objects and other contexts.

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Homomorphisms do a lot of things:

1. They can be used to show two structures are identical (isomorphism)

2. They can be used to show one structure is a substructure of another (monomorphism)

3. They can be used to show one structure is a quotient of another (epimorphism)

4. General homomorphisms are kind of a mixture of the above, carrying a certain amount of data about the relationship between the domain and codomain.

This list is not exhaustive, it's just meant to sample some of what homomorphisms do.

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Any homomorphism of [groups, rings, modules] is an epimorphism followed by a monomorphism, so it is a precise mixture of the above. In general, I think any morphism in an abelian category is an epi followed by a mono, but I don't know if this condition on the category is necessary or not. –  M Turgeon Jul 5 '12 at 17:39

Groups, rings (especially fields) and many other objects in mathematics form so called categories. I.e. the collection of all rings (whatever this is) forms a category. To 'compare' the objects in a category, one needs morphisms between them. (This is not the only need for morphisms, but one of them). To say that one object is 'bigger' or that two objects are isomorphic, one needs the notion of morphism.

Morphisms are generally maps (though not all morphisms need to be maps) that respect the structure of the objects.

Of course, morphisms have other useful applications. Matrices for example (i.e. morphisms of vector spaces [or modules]) are connected to linear equations and so on...

Morphisms play very many very important roles in mathematics :)

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Another function of morphisms is as a way of taking an unfamiliar or apparently complex algebraic object, and placing it in a more familiar context, or one where there are known analytical tools and methods - as in representation theory.

The point of view you take depends rather on whether you are more interested in studying particular objects, or alternatively classes of objects.

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Let's suppose for the moment that you really only care about isomorphisms. That is, you really only care about telling whether two groups are isomorphic, or something. Great! But how do you write down isomorphisms? It's hard. Often you will be able to write down a map and you'll want to show it's an isomorphism. Before you show it's an isomorphism, what is it?

It's probably a homomorphism!

So even if you only care about isomorphisms, in the course of proving things about isomorphisms you will very likely end up having to talk about homomorphisms which are not necessarily isomorphisms (at least you don't know that they are), so you might as well talk about them from the outset. Moreover,

• you can use homomorphisms to construct isomorphisms,
• even homomorphisms which are not isomorphisms can be used to prove useful statements, e.g. that there do not exist certain isomorphisms,
• etc.

But really to understand this you should do a lot of mathematics. It will become clearer to you as you gain experience with working with maps.

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+1 for "even homomorphisms which are not isomorphisms can be used to prove useful statements" –  user3533 Jul 5 '12 at 21:31

I think the use of homomorphisms is to identify one algebraic structure with another. With that identification we can say that two algebraic structures are same structurally i.e they are same when we view them with respect to algebraic operations involved in them although they may seem different with open eye.

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So all groups are the same with respect to algebraic operations to the trivial group, due to the trivial homomorphism? I think you are talking about isomorphisms rather than homomorphisms. –  Alex Becker Jul 5 '12 at 16:56
@AlexBecker: Yes I am talking about isomorphisms. I think the major use of homomorphisms is through isomorphisms. –  pritam Jul 5 '12 at 17:25
Certainly not. Almost all homomorphisms I work with are not isomorphisms. Homomorphisms which are not isomorphisms are hugely important, such as in homology and cohomology theories. –  Alex Becker Jul 5 '12 at 17:30
@AlexBecker: Sorry, I did not know that, I have not gone that far in algebra. –  pritam Jul 5 '12 at 17:39