Can $(R,.,+)$ forms a ring

Question

From the definition of a ring we know that $~(R,+,.)~$ is ring if $~(R,+)~$ is an abelian group and $~(R,.)~$ is a semi-group, that is closed and associative and multiplication is distributive. Now my question, is there any ring exists of the form $~(R,.,+)~$?
That is, let $~R=\mathbb R-\{0\}.~~$ Then $R$ is not closed under addition. In this case $(R,.,+)$ cannot be a ring. So can there any example such that $(R,.,+)$ forms a ring? Where multiplication and addition has same definition.
For jist, can we define a ring $(R, \oplus, \odot)$ such that $$a \oplus b =a.b ~~\text{and}~~~ a \odot b=a+b.$$

Answer 1

Yes. Consider the zero ring (the ring with exactly 1 element).

Note that any ring $(R, +, \cdot)$ such that $(R, \cdot, +)$ must be the zero ring, since $0$ would have to be a unit. Whenever $0$ is a unit, we have $0 \cdot a = 1$ for some $a$. But $0 \cdot a = 0$, and hence $0 = 1$, and hence the ring is the zero ring.