Armendariz ring: $R$ is said to be an Armendariz ring if whenever $f(x)= \displaystyle\sum_{i=0}^{n}a_ix^i$ and $g(x)=\displaystyle\sum_{j=0}^{m}b_jx^j \in R[x]$ satisfy $f(x)g(x)=0$, then $ a_ib_j=0$ for each $i,j$.
Tripotent ring: A ring $R$ is tripotent if its each element is tripotent i.e $x^3=x$ for all $x\in R$.
My attempt: For example $\mathbb{Z}_3$ is tripotent but $\mathbb{Z}_n$ is Armendariz for every positive integer $n$ by this paper.
Some interesting results on tripotent rings are:
(1) Every element of tripotent ring can be written as a sum of two commuting idempotents.
(2) If $R$ is a commutative ring in which every element is the sum of two idempotents then $R$ is a tripotent ring.
