# Rings with zero divisors

Is there a ring $~R~$ with non-trivial multiplication (i.e. $~\exists a,b\in R ~~~ ab\neq 0$) such that each non-zero element of $~R~$ is a zero-divisor?

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Presumably, then, you allow rings without identity, since the identity cannot be a zero divisor. –  Thomas Andrews May 20 '13 at 14:00
@ThomasAndrews I use the following definition of a ring: (R,+) is an abelian group; multiplication is associative and distributive. Multiplication identity isn't required. –  Igor May 20 '13 at 14:03

The simplest example might be $R=2\mathbb Z/8\mathbb Z$. Then $2\cdot 2\neq 0$ but $a\cdot 4=0$ for all $a\in R$.
A (trivialish) example is $\{0, 1\}$ with $1 \cdot 1 = 0$.