# How to find zerodivisors in finite commutative rings

15. December 2020 ‐ **1 min read**

Given a question similar to

Find all zerodivisors in the ring $\mathbb{Z}_{25}.$

How do we solve this task efficiently? First, we can recall that in a finite commutative ring all elements are either units or zerodivisors. Now, an element $a$ of the ring is a unit if and only if $\gcd(a,m) = 1$ (in the previously mentioned example, $m$ would be 25).

Therefore all elements $a$ where $\gcd(a,m) \ne 1$ are zerodivisors. Phrased differently, this means that all elements that are **not** relatively prime to $m$ are zerodivisors of $\mathbb{Z}_m$.

Applying this to our example from the start, we want to find all the elements in the ring $\mathbb{Z}_{25}$ that are not relatively prime to $25$. Since $25 = 5 \times 5$, those elements are exactly the multiples of $5$, so the solution is $$\lbrace 5, 10, 15, 20 \rbrace$$. ($25 = 0$ and $0$ is never a zerodivisor by definition).