Question

If `(R,*_1,*_2)` is a ring, then `ψ:R×R->R:ψ(r_1,r_2)=r_1*_2r_1` is a binary operation. This statement is true/false? Please give reasons your answer ?

Answer 1

I'd say yes

Explanation:

It comes to what is exactly your definition of binary operation. I gave a quick lookup online, and wikipedia defines a binary operation as a function `f` whose two domain and codomain are the same set: `f: S \times S \to S`.

Your application indeed maps `R \times R` unto `R`, and we are guaranteed that `r_1 \cdot_2 r_2 \in R`, because `R` is a ring and is thus closed with respect to the operation `\cdot_2`.

So, unless there are other requirements for an application to be a binary operation, I'd say that `\psi` meets all the requirements.