Question

Prove that a ring with one consists only of zero if and only if 1 = 0

Answer 1

($\Longleftarrow$) Suppose that $1=0$ in ring $R$ with $1$.

For any element $a\in R$ using the property $a\cdot0=0$ we have: $a=a\cdot1=a\cdot0=0$.

So, $R=\{0\}$.

($\Longrightarrow$) If $R=\{0\}$ it is obvious that $1=0$.