"distributive property" vs. "ring homomorphism": comparing definitions

Question

The property of distributivity is defined using expressions like the following, from page one of "Introduction to Commutative Algebra" by Atiyah and MacDonald:

$$x(y + z) = xy + xz$$ $$(y + z)x = yx + zx.$$

On the other hand, homomorphisms are defined using expressions like the following, from the definition of ring homomorphism on page two of Atiyah-MacDonald:

$$f(x + y) = f(x) + f(y)$$

These expressions obviously bear some resemblance to one another, when viewed simply as strings of characters, but does this observation lead to any interesting examples or generalizations? Can the definitions be restated so as to have one subsuming the other?

Answer 1

This is a good observation. The key here is that we want multiplication and our homomorphisms to be compatible with our addition operation. When we study groups, we have a set $G$ equipped with a single binary operation $G\times G\to G$ satisfying elementary properties: associativity, existence of inverses, and existence of an identity element. When we study rings $A$, we have two operations $+$ and $\times$, namely addition and multiplication. Usually $(A,+,\times)$ is required to be an Abelian group under $+$ and a (often commutative) monoid under $\times$ (inverses might not exist under multiplication, but the other properties of a group are respected).

The natural question remains: how should the operations interact with each other? If there is no interaction between the multiplicative and additive structures of the ring, we might as well separately study $(A,+)$ and $(A,\times)$ as an Abelian group and a monoid, respectively. So, we require that $\times$ "preserve" the group structure of $(A,+)$. A sensible interpretation of this is the following: the map $\phi: A\times A\to A$ given by $(x,y)\mapsto x\times y$ should be "like a homomorphism of $(A,+)$ to itself." With suitable modification, it is. If we fix $x$, then the map $$ \phi_x:(A,+)\to (A,+)$$ has $$ \phi_x(y+z)=\phi(x,y+z)=x(y+z)=xy+xz=\phi_x(y)+\phi_x(z)$$ and if we fix $y$, the map $$ \phi^y:(A,+)\to (A,+)$$ has $$ \phi^y(x+z)=\phi(x+z,y)=(x+z)y=xy+zy=\phi^y(x)+\phi^y(z).$$ So we have a pair of group homomorphisms associated with right and left multiplication.

Answer 2

Well, you could word the left distributive law say that for each $x$, the operator $f_x$ defined by the formula $f_x(y)=xy$ is a homomorphism of the additive structure.

Answer 3

I do not think it is a mere coincidence.

Clearly the operation of addition is structural in a ring. What I mean by this is that it is a part of the ring structure by definition. So it seems natural that the functions, operations, constructions, etc., one usually considers in the Ring theory are "compatible" with this structure.

Next, I would argue that the compatibility condition like this $$ f(x+y)=f(x)+f(y) $$ is the most natural. Think about it this way: the image of a sum is the sum of images, so the sum in the 1-st ring should go to the sum in the 2-nd ring.

Now the two distributive laws: $$ x(y+z)=xy+xz \ \mbox{ and } \ (y+z)x=yx+zx, $$ can be interpreted as follows. The function of the left multiplication by $x$, $L_x(r)=xr$, and the function of the right multiplication by $x$, $R_x(r)=rx$, are compatible with the addition as described above.

At the end I want to mention that the other explanation for axioms of any abstract algebraic concept (group, ring, field, homomorphism, ideal) is that there are several important examples satisfying these axioms.