Prove Isomorphism: R x S & S x R

[stripes]

Homework Statement

Show that for any rings R and S, R x S and S x R are isomorphic, and A x B is the cartesian product, or ordered pairs. So an element of R x S can be written as (r₁, s₁).

Homework Equations

The Attempt at a Solution

So I have to show that there is a bijection from R x S to S x R, and this bijection must preserve addition and multiplication. This is tough for me since the mapping from R x S to S x R could be anything! How can I even start if I don't have this function or mapping?

 

[tiny-tim]

hi stripes!

… the mapping from R x S to S x R could be anything!

yes it could

but you're in charge, and you can choose any mapping you like

go for the "natural" mapping …

what do you think it would be really neat for (r₁, s₁) to be mapped onto? ​