I found a linear function $f$ with "weird" multiplication $f(a)f(b)=f(af(b))=af(f(b))$. Is it of any use?

Question

I found a function $f:Z \rightarrow Q$ (the domain may be extended to $Q$) such that, if $a+b=c \in Z$, we can perform $f(a)+f(b)=f(c)$, and this function has the properties:

1. $xf(a)+yf(b)=f(xa+yb)$ (very similar to Z)

2. $f(a)f(b)=f(af(b))=af(f(b))$ (a weird multiplication)

3. $nf(x)=f(nx)$ (homogeneous)

Question Can you find any good use of this?

Answer 1

By property 3 alone, we already have $f\left(n\right)=f\left(n*1\right)=nf\left(1\right)$, so this map just scales the integers (and possibly flips them if $f(1)<0$). Properties 0. (the $f(a)+f(b)=f(c)$ thing) and 1. and 2. all hold automatically.

There are tons of things to learn in mathematics that build upon examples like this one, so I encourage you to keep thinking and also to keep reading/studying.

For instance, your function is a linear map from the integers to the rationals over the integers (to anyone worried that the integers aren't a field, they're still a ring so we have a module). It's therefore also a group homomorphism from the integers to the additive group of the rationals.