Is there motivation for a ring with operations $x\oplus y = x+y-u$ and $x\odot y=x+y-u^{-1}xy$? This question is a basic exercise in verifying whether something is a ring or not. In a slightly generalized version, the result is the following:

Let $(R,+,\cdot)$ be a ring, and let $u\in R^*$ be a fixed invertible element. Then $\tilde R=(R,\oplus,\odot)$ with
$$x\oplus y = x+y-u, \quad x\odot y=x+y-u^{-1}xy$$ is a ring (with $\tilde0=u$, $\tilde1=0$). Also note that $\tilde R$ inherits (at least) commutativity, "integral domain"-ness and field-ness from $R$.

I got curious: Is there any nice motivation for the construction? Is this a natural thing to do/study?
I did note that we can rewrite
$$
x\odot y = -u^{-1}(x-u)(y-u) + u,
$$
which has a nice symmetry to it. Perhaps a geometric interpretation is possible? The sum at least can be seen as the regular sum with the "origin" shifted to $u/2$.
 A: This is nothing more than transport of structure, which can be done with any bijection from $R$ to another set.  (The wiki article doesn't seem to list transport of algebraic operations as an example right now, but it is indeed called this in practice.)
That is, given a ring $R$, a set $X$ and a bijection $R\to X$, we may declare new operations $\oplus, \otimes$ on $X$ given by
$$
f(x)\oplus f(y):=f(x+y)\\
f(x)\otimes f(y) := f(xy)
$$
and $(X, \oplus,\otimes)$ becomes a ring. Moreover, $f$ is a ring isomorphism between the two.
That is what is happening here, where we are using the bijection $x\mapsto -ux+u$. Since $u$ is assumed invertible (otherwise $u^{-1}$ has no meaning) the assignment is a bijection.
So yes, your intuition that there's some geometric intuition is justified since $-ux+u$ is a scaling followed by a translation inside $R$. The bijection is built purely out of ring operations. But keep in mind that depending on $R$, scaling and translating might be a lot weirder than it is in a nice ordered field like $\mathbb R$.
When you use an arbitrary bijection $f$ I wouldn't say there is any geometric intuition, since the rearrangement could be quite arbitrary.  You could call that geometric but it seems to be more combinatoric at that point.
