Exercise 3.5

Let $R$ be a ring, and assume that $0=1$ in $R$. Prove that $R$ consists of a single element.

Answer

Since $0$ is in $R$, it is nonempty and thus has at least one element. We will prove that it has at most one element. Let $a\in R$. Then since $0=1$, $a=a\cdot 1=a\cdot 0=0$. Thus, every element of $R$ is $0$. Therefore, $R$ only consists of a single element. $■$