How to construct a ring isomorphism in this case?

Question

Let (A, +, $\times$) be a ring with unity 1 and (A, $\oplus$, $\otimes$) be a ring (already proved to be), where the operations are defined as:

• a $\oplus$ b = a + b + 1

• a $\otimes$ b = (a $\times$ b) + a + b

We want to formally verify if (A, +, $\times$) and (A, $\oplus$, $\otimes$) are isomorphic.

My question is: is there a path to do is? I know that I must find a bijective homomorphism, but I don't know any ideas to construct such function given these rings. I tried but can't find any promising ideas, so any general light on this type of problem would be of great value.

Answer 1

The comment of @dan_fuela above gave me an insight and I got to answer the question, so I am writing it down here.

Seeing

• (a $\oplus$ b) as (a + 1) + (b + 1) + (-1), and
• (a $\otimes$ b) as (a + 1) $\times$ (b + 1) + (-1)

we are inspired to define f: A $\rightarrow$ A as f(a) = a + (-1). And indeed, it is easy to follow that f(a + b) = f(a) $\oplus$ f(b) and that f(a $\times$ b) = f(a) $\otimes$ f(b).

Moreover, if f(a) = f(b), then a + (-1) = b + (-1), so a = b. Thus f is injective. And if b $\in$ A, we can take a $:=$b+1 so f(a) = a + (-1) = b. Thus f is surjective.

Therefore, f is bijective and we got ourselves an isomorphism.