If every element of R has a (multiplicative) inverse, then $R = \{0\}$

We have $0 \cdot x = 1$ for some $x \in R$, so $1 = 0$, and it follows that for $x\in R$, $x = x\cdot 1 = x \cdot 0 = 0$.

First of all if every non zero element of $R$ has a multiplicative inverse that makes it a division ring and not a field a good example is the quaternions it only becomes a field if the multiplication is commutative. Secondly from the axioms of ring theory $0$ does not have a multiplicative inverse except for the ring {$0$}