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?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
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 $a\in R$.
To get a general class of examples, you can take the nilradical of any commutative ring (ie, the set of nilpotent elements). This forms an ideal consisting solely of nilpotent elements (and hence of zero-divisors), but will usually have non-trivial multiplication.