Left Derivations and Strong Commutativity Preserving Maps on Semiprime $\mathrm{\Gamma }$-Rings

In this paper, firstly as a short note, we prove that a left derivation of a semiprime $\mathrm{\Gamma }$-ring $M$ must map $M$ into its center, which improves a result by Paul and Halder and some results by Asci and Ceran. Also we prove that a semiprime $\mathrm{\Gamma }$-ring with a strong commutativity preserving derivation on itself must be commutative and that a strong commutativity preserving endomorphism on a semiprime $\mathrm{\Gamma }$-ring $M$ must have the form $\sigma (x)=x+\zeta (x)$ where $\zeta$ is a map from $M$ into its center, which extends some results by Bell and Daif to semiprime $\mathrm{\Gamma }$-rings.