In this paper, the authors introduce a new notion of generalized (Jordan) left derivation on rings as follows; let \(R\) be a ring, an additive mapping \(F: R\to R\) is called a generalized (resp., Jordan) left derivation if there exists an element \(w\in R\) such that \(F(xy) = xF(y) + yF(x) + yxw\) (resp., \(F(x^2) = 2xF(x) + x^2w)\) for all \(x,y\in R\). In addition, some related properties and results on generalized (Jordan) left derivation of square closed Lie ideals are obtained.
In fact, the authors focus on Lemma 2.1, which contains 6 parts, but they leave it without proof. I believe the paper would be better if they would have provide at least two of these parts. However, the authors mention good background information. To become more closely to definition of generalized (Jordan) left derivations on rings the authors supply some examples to illustrate it. In the proof of Theorem 2.1, I believe something is not completely correct, since the authors rely on Lemma 2.1, stating that \((F,w):R\to R\) is a generalized Jordan left derivation on a square closed Lie ideal of \(R\), while Theorem 2.1 mentions that \((F,w):R\to R\) is a generalized Jordan left derivation on a 2-torsion free semiprime ring \(R\). The authors apply this result of the proof for another theorems.