Let $R$ be a prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C=Z(U)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$, $F$ a non-zero generalized derivation of $R$. Suppose that $[F(u),u]F(u)=0$ for all $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F(x)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity $s_4$ and there exist $a\in U$ and $\alpha \in C$ such that $F(x)=ax+xa+\alpha x$ for all $x\in R$. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.
The rock units of the NW Himalayan region are fragile, heavily fractured and highly deformed due to active tectonics and complex geological setup. Fast urbanization, road constructions along hill slopes and other infrastructural development activities also increased the slopes instability problems. The present study emphasizes the application of rock mass classification to estimate the rock mass properties along the Yadgar section Muzaffarabad, NW Himalayas, Pakistan. For this purpose, Rock Mass Rating (RMR) and Geological Strength Index (GSI) were used to characterize and classify the rock masses. In the present study, twenty-five sites have been investigated to evaluate rock properties along the Muzaffarabad-Neelum road, Sub-Himalayas, Pakistan. Result of the study shows that the Abbottabad Formation of Cambrian age is vulnerable in the Yadgar section with extremely poor RQD (Rock Quality Designation), lowest UCS (Unconfined Compression Strength) values and closely spaced discontinuities. and RMR values of the Abbottabad Formation ranges from 40-54 and classified as Poor to Fair having low GSI (20±3-35±3), blocky, disintegrated structure. The Paleocene Hangu Formation has lowest GSI (28±3-29±3; Blocky, Disturbed/ Seamy in nature) having RMR (40-45) and Eocene Kuldana Formation has GSI (30±3-45±3; Blocky) having RMR (34-67), are categorized as heavily broken, disintegrated and poorly interlocked rock masses. RMR values of rock units of the Paleocene Lockhart Formation (52-60), the Miocene Murree Formation (38-63), and the Eocene Margala Hill Limestone (38-61) are relatively higher values having GSI values ranges from (35±3-45±3; 35±3-50±3; 30±3-40±3) res)ectively. RMR and GSI values in Yadgar section, ranges between 34-67 and 20±3-50±3 respectively. Analysis shows positive correlation between GSI and RMR values. This approach to evaluate the rock mass classification through RMR and GSI will give the better estimation of rock mass properties along Muzaffarabad-Neelum road to identify the vulnerable slopes and design effective geotechnical measures.