Fields and Groups equivalent

Question

Are the following two statements equivalent to each other ?

1) $(R,+,*)$ is a field

2) $(R,+)$ is an abelian group and $(R\setminus\{0_R\},*)$ is an abelian group

If not give an example.

Answer 1

No. There is one more property satisfied in a field: the distributivity between $+$ and $\times$.

If the distributivity is satisfied, then $A$ is a ring in which non zero elements are $\times$-invertible. It is, indeed, a field.