Let $K$ be an assotiative ring with 1 nonzero multiplication. Is it true that if $K$ consists of two elements then $K \cong \mathbb{Z}_2?$
It is clear that second element is $0$ and $1 \cdot 1=1, 1\cdot 0=0, 1+0=1, 0+0=0$ but what about $1+1?$ It must be equal only $0$ and $1+1=1$ is impossyble?
$1+1=1$ is not possible, since $1+1=1\Longrightarrow (1+1)-1=1-1\Longrightarrow$1=0.
Yes, it must be isomorphic to $Z_2$.
For if $1 + 1 = 1$, you can add $-1$ to both sides to get $-1 + 1 + 1= -1 + 1$, so $0+1= 0$ so $1 = 0$, and you don't have a 2-element ring, because your two elements are the same.
