Can there be different ring structures on the same set? Say we have the set of real numbers. Can we construct a different ring than the usual ring of real numbers?
I am trying to wrap my head around the idea of rings and I couldn't find two other operations that will hold distributive property other than usual addition and multiplication.
If we can find another ring with the set of real numbers, is there a set with only one ring structure? If we can't find another ring with the set of real numbers, is there a set with different ring structures?
 A: Here is a more explicit example if my comment about $\Bbb Z$ and $\Bbb Q$ was unconvincing.
Let $S=\{a,b,c,d\}$. Let $R_1$ be the ring structure on $S$ inherited from $\Bbb Z/(2)\times\Bbb Z/(2)$ under the bijection $a\mapsto (0,0)$, $b\mapsto (1,0)$, $c\mapsto (0,1)$, and $d\mapsto (1,1)$. For example, $ab=a$, $b+d=c$, and $cd=c$.
Now let $R_2$ be the ring on $S$ inherited from $\Bbb Z/(4)$ with the bijection $a\mapsto 0$, $b\mapsto 1$, $c\mapsto 2$, and $d\mapsto 3$.
You can verify that $R_1$ and $R_2$ are not isomorphic since, for example, $R_1$  has no element of (additive) order $4$.
A: The other examples given are nice, but to give an example where the set is $\mathbb{R}$ as you were originally wondering about you take $\mathbb{R}$ with the normal multiplication but where we define the multiplication operation to always evaluate to $0$. Under this definition we have a different ring structure and it's certainly not isomorphic since this structure is not a field. This isn't a particularly nice ring since it doesn't have unity.
A: Given any bijection $\,f:\mathbb{R}\to\mathbb{R},\,$ define new addition and
multiplication operations by $\,x\oplus y:=f^{-1}(f(x)+f(y))\,$ and
$\,x\otimes y:=f^{-1}(f(x)\cdot f(y)).\,$ These two operations form
$\,\mathbb{R}\,$ into a ring isomorphic the the usual ring of reals.
As a concrete example, try $\,f(x):=x+1.\,$ This example is given in an answer to
MSE question 1911294
"Operations on the vector set $\mathbb{R}$ that will provide a vector space".
