# Ring where all elements are zero divisors.

I know that the collection of all nilpotent elements in a commutative ring form such a ring where all elements are zero divisors. For example, if we take $$S=\lbrace 0,2,4,6 \rbrace$$ as a subring of $$\mathbb{Z_8}$$, then all elements of $$S$$ are zero divisors and nilpotent. If we look at $$\mathbb{Z_{p^n}}$$, then we arrive at another example of this. But I'm struggling to find a well-known example where not every element is nilpotent. More specifically, I want to find such a ring that at least one of the element of the ring is a zero divisor but not nilpotent. How should I think of such an example?

• I gather that by "ring" you mean ring without 1. Commented May 24 at 19:49
• No it may have a unity Commented May 24 at 19:50
• Well $1$ can never be a zero divisor. Commented May 24 at 19:53
• " I want to find such a ring that atleast one of the element of the ring is a zero divisor but not nilpotent." - split-complex numbers. Commented May 25 at 14:28

## 3 Answers

What about $$\{0,3,6,9,12,15\}$$ where addition and multiplication is $$\pmod{18}$$?

• This should work. Commented May 24 at 20:06
• or $\{0,2,4,6,8,10\}$ $\pmod{12}$? Commented May 24 at 21:47
• More generally, $k\mathbb{Z}/n\mathbb{Z}\subset\mathbb{Z}/n\mathbb{Z}$ (where "$k$" divides "$n$") will always give an example of a finite ring in which every element is a zero divisor, as well -- in most cases -- where no elements are nilpotent. Commented May 28 at 21:10
• @JAG131: if $n=k^2$, all the elements of $k\mathbb Z/n\mathbb Z$ are nilpotent; if $n=kl$, where $k$ and $l$ are distinct primes, then $k\mathbb Z/n\mathbb Z$ does not have nonzero zero divisors Commented May 28 at 21:31
• @JAG131: such as $n=k^2l$ with $\gcd(k,l)=1$ Commented May 28 at 21:48

I think this ring will work as well $$R=\Bigg\lbrace \begin{pmatrix} 0& a \\ 0 & b \\ \end{pmatrix}|a,b \in \mathbb{Z} \Bigg\rbrace$$.

The residues $$0,6,10,15\bmod 30$$ form a multiplicative ring in which every element is idempotent while the product of any two distinct elements is $$0$$.

• This is an interesting ring. Commented Jun 3 at 19:56