Among the results of the paper is the following proposition. Let $(Q,A,M)$ be an $(m,n)$-ring and let $\mathbf O$ the $\{1,m\}$-neutral operation of the $m$-group $(Q,A)$. Then for every $i\in\{1,\ldots,n\}$ and for every $a^{n-1}_1,c^{n-1}_1\in Q$ the following equality holds \[ M(a^{i-1}_1,\mathbf O(c^{m-2}_1),a^{i-1}_i)=\mathbf Overline{(M(a^{i-1}_1,c_j,a^{n-1}_i)}|^{m-2}_{j=1}). \]