  Let $\mathcal R$ be a finite valuation ring of order $q^r$ with $q$ a power
of an odd prime number, and $\mathcal A$ be a set in $\mathcal R$. In this
paper, we improve a recent result due to Yazici (2018) on a sum-product type
problem. More precisely, we will prove that
  1. If $|\mathcal A|\gg q^{r-\frac{1}{3}}$, then $$\max\left\lbrace |\mathcal
A+\mathcal A|, |\mathcal A^2+\mathcal A^2|\right\rbrace \gg
q^{\frac{r}{2}}|\mathcal A|^{\frac{1}{2}}.$$ 2. If $q^{r-\frac{3}{8}}\ll
|\mathcal A|\ll q^{r-\frac{1}{3}}$, then $$\max\left\lbrace |\mathcal
A+\mathcal A|, |\mathcal A^2+\mathcal A^2|\right\rbrace \gg \frac{|\mathcal
A|^2}{q^{\frac{2r-1}{2}}}.$$ 3. If $|\mathcal A+\mathcal A||\mathcal A|^2\gg
q^{3r-1}$ and $2q^{r-1}\le |\mathcal A|\ll q^{r-\frac{3}{8}}$, then
$$\max\left\lbrace |\mathcal A+\mathcal A|, |\mathcal A^2+\mathcal
A^2|\right\rbrace \gg q^{r/3}|\mathcal A|^{2/3}.$$
