Let $S$ a non-empty set, $(A,+,\cdot)$ a ring and $f:S\to A$ a bijection. Defined the operations $xy$ and $x\,\Delta\,y$, is $(S,,\Delta)$ a ring?

25 Views Asked by Bumbble Comm At 29 Mar 2026 - 4:32

Let $S$ a non-empty set, $(A,+,\cdot)$ a ring and $f:S\to A$ a bijection. To $x,y\in S$, we define the operations this way:

$x*y=f^{-1}(f(x)+f(y))$ and $x\,\Delta\,y=f^{-1}(f(x)f(y))$.

Verify that $(S,*,\Delta)$ is a ring.

I'm having trouble to deal with those functions inverted.

In the association's proof, we have $(x*y)*z=[f^{-1}(f(x)+f(y)]*z=f^{-1}(f^{-1}(f(x)+f(y))+f(z))$

and

$x*(y*z)=x*[f^{-1}(f(y)+f(z))]=f^{-1}(f(x)+f^{-1}(f(y)+f(z)))$

So, I think I need to show that $$f^{-1}(f^{-1}(f(x)+f(y))+f(z))=f^{-1}(f(x)+f^{-1}(f(y)+f(z)))$$ and i don't know how to do it.

Could you guys give me a hint?

Let $S$ a non-empty set, $(A,+,\cdot)$ a ring and $f:S\to A$ a bijection. Defined the operations $x*y$ and $x\,\Delta\,y$, is $(S,*,\Delta)$ a ring?