Why is homomorphism defined as such?

Question

I am studying ring homomorphism and was curious why it is defined as such.The definition I have is

Let $(R,+,*)$ and $(S,\oplus,\cdot)$ be two rings. A map $\varphi : R \to S$ is called a homomorphism if for every $a,b \in R, \varphi(a+b)= \varphi(a) \oplus \varphi(b)$ and $\varphi(a*b)=\varphi(a)\cdot\varphi(β)$

I get that since $a,b\in R$, then $a * b\in R$ as well, and that $\varphi(a),\varphi(b),\varphi(a*b)\in S$, as well as that $\varphi(a)\cdot\varphi(b)\in R$. But why does $\varphi(a*b)=\varphi(a)\cdot\varphi(b)$ though?

Thanks in advance :)

Answer 1

Here's a slightly different way of getting an intuition about this. Let's say we did have a bijection $\varphi:R\to S$. Then the most natural way of "mapping" the operations would be to say $s\oplus s' = \varphi(\varphi^{-1}(s)+\varphi^{-1}(s'))$ and similarly for the multiplicative structure. This can be equivalently written $r+r'=\varphi^{-1}(\varphi(r)\oplus\varphi(r'))$. Finally, this can also equivalently be written $\varphi(r+r')=\varphi(r)\oplus\varphi(r')$. This last form no longer requires assuming that $\varphi$ is a bijection.

As a historical note, quoting from the second footnote of Colin McLarty's The Uses and Abuses of the History of Topos Theory (PDF):

Group theorists into the 1950s generally counted as homomorphisms only isomorphisms and projections onto quotient groups. In hindsight they lacked the idea of a codomain as opposed to an image so they recognized only surjective homomorphisms, where the image coincides with the codomain.

I suspect that the way this manifested is they wouldn't talk about a homomorphism between groups, but rather the group structure induced by a bijection via my first two equations in the case of isomorphism. For projections onto quotient groups, this just looks like considering the "same" operation only "mod $N$" (for $N$ a normal subgroup).

Answer 2

A ring is a set with some extra structure. It is a general theme in mathematics: whenever we have sets with some extra structure we want to define maps between them so that they preserve that structure.

If we try to define what it means "to preserve structure" in the case of rings, then the only natural way to do so is as follows.

Given rings $A$ and $B$, and a map $\varphi\colon A\to B$ we want the following two procedures to yield the same result:

1. Take two elements in $A$, add (or multiply) them and apply $\varphi$ to the result.
2. Take two elements in $A$, apply $\varphi$ to both of them and THEN add (or multiply) them in $B$.

This is essentially the definition of a homomorphism of rings.