Zero divisor in ring without unity

Question

Let $R$ be a commutative ring without unity and $n \in R\setminus\{0\}$. Prove that $n\mid n$ implies that $n$ is a zero divisor.

Answer 1

Assume that $n=nk$ for some $k\in R$. Because $R$ has no unity, there exists an element $r\in R$ such that $kr\neq r$. In other words $kr-r\neq0$.

But by standard applications of rng axioms $$ n(kr-r)=n(kr)-nr=(nk)r-nr=nr-nr=0. $$ Therefore $n$ is a zero-divisor.