Example of ring $R$ when $(R,.)$ forms a group

Question

If $(R,+,.)$ is a ring and $(R,.)$ is monoid then is $(R,.)$ is a group or not?

This statment is false in general as,it is not necessary that multiplicative inverse of each element exists,as an example multiplicative inverse of 2 does not exist in ($\mathbb Z$,.)

I need an example of $R$ when $(R,.)$ forms a group

Answer 1

One of the first properties of rings that are derived from the axioms is $$ r0=0 $$ for every $r\in R$.

Suppose $0$ has an inverse, $0^{-1}$. Then, for every $r\in R$, $$ r=r1=r(0^{-1}0)=(r0^{-1})0=0 $$ so the only element of $R$ is $0$.