TITLE: What is the significance of multiplication (as distinct from addition) in algebra & ring theory?
QUESTION [10 upvotes]: In higher math, operators are defined over a set of objects; and these operators are usually denoted as addition and multiplication with a distribution rule.  Assuming multiplication is not repeated addition -- though it can be thought that way in some contexts -- what are the needs that push us to define two operators like this (say, if we hadn't learned the usual multiplication already)?  
Why not ten operations?  Does anything more complicated always reduce to just two operators?
In some areas, multiplication is sometimes a cross product or even a bracket/commutator.  Is there some underlying idea here that "multiplication" represents?  If so, what is the principle that causes us to generalize?

REPLY [4 votes]: Widespread utility: many interesting and useful structures have operations satisfying the group or ring axioms, and there are a lot of useful techniques for studying them, especially for special classes of them.
Sometimes, it is even useful to invent a group or ring to describe a situation where no such structure already exists, simply because we know how to study them.
In other words, we study ring theory not because there is some deep idea that all algebraic structures are groups and rings, but because there is a lot that we can say about them, and we can use that knowledge a lot.

REPLY [3 votes]: Whenever a binary operation is associative, whether it is commutative or not, it can be "represented" as an operator, with the binary operation corresponding to "composition."
For example, let $(G,\times)$ be associative. For each $g\in G$, define $\phi_{g}:G\to G$ as $\phi_g(h)=g\times h$.  Associativity can then be written as:$$\phi_g\circ \phi_h = \phi_{g\times h}$$.
Indeed, in general, $(X,\times)$ is associative if and only if the map $X\to X^X$ taking $x\to \phi_x$ is a homomorphism of $(X,\times)$ and $(X^X,\circ)$ as sets with binary operators.
It is because of this relationship between associative operations and function composition that we often write binary operations as multiplication - the fact that we often write $\phi_1\phi_2$ for $\phi_1\circ\phi_2$.
This "representation:" $$(X,\times)\to(X^X,\circ)$$ is not necessarily faithful. It is possible for $\phi_x=\phi_y$ with $x\neq y$. But most of the cases of associative operators that you run into in the real world are faithful under this representation. In particular, whenever the binary operation has an identity element, this representation is faithful.
Indeed, we often get restrictions. Multiplication in a ring, for example, has the distributive law, which essentially means that the $\phi_r$ are homomorphisms of the abelian group $(R,+)$.  So here, the representation is $(R,\times)\to (Hom_{\mathcal {Ab}}(R,R),\circ)$. Or for groups, $\phi_{g}$ are always $1-1$ and onto. That "subset" relationship - what collection of "maps" from $X$ to $X$ do we allow for the $\phi_x$ - is often the defining characteristic of our class of algebras.