Abstract. The main purpose of this article is to prove the following result: For integers $m$, $n$ with $m \geq 0$, $n \geq 0$, and $m + n \neq 0$, let $\mathcal {R}$ be an $(m + n + 2)!$-torsion free prime ring with the identity element $\textbf {e}$. Suppose that $d, \sigma \colon \mathcal {R} \rightarrow \mathcal {R}$ are two additive mappings such that $\sigma $ is a monomorphism with $\sigma (\textbf {e}) = \textbf {e}$, and $d(\mathcal {R}) \subseteq \sigma (\mathcal {R})$. If $d$ and $\sigma $ satisfy both of the equations \[d(xy)(\sigma (z)-z)-d(x)(\sigma (yz)-\sigma (y) z)+\sigma (xy) d(z)-\sigma (x)(d(yz)-d(y)z) = 0\] and \[d(x^{m + n + 1}) = (m + n + 1)\sigma (x^m) d(x) \sigma (x^n)\]for all $x, y, z \in \mathcal {R}$, then $d$ is a $\sigma $-derivation.

DOI: 10.14232/actasm-018-594-6

AMS Subject Classification (1991): 47B47; 16N60

Keyword(s): derivation, $\sigma $-derivation, 2-torsion free prime ring, two-variable $\sigma $-derivation, commutativity of rings

Received November 15, 2018 and in final form May 11, 2019. (Registered under 94/2018.)