In this paper, we investigate the commutativity of a semiprime ring R admitting a generalized derivation F with associated derivation D satisfying any one of the properties: (i) F (x) • D(y) = [x, y], (ii) D(x) • F (y) = F [x, y], (iii) D(x) • F (y) = xy, (iv) F (x • y) = [F (x), y] + [D(y), x], and (v) F [x, y] = F (x) • y − D(y) • x for all x, y in some appropriate subsets of R.

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The purpose of this paper is to prove some results which are of independent interest and related to additive maps on σ-prime rings. Further, examples are given to demonstrate that the restrictions imposed on the hypotheses of these results are not superfluous.