Let $\mathcal{N}$ be a nest on a Banach space $\mathcal{X}$ and $\mathcal{A}$ be a standard subalgebra of $Alg\mathcal{N}$. Let $\phi: \mathcal{A}\times \mathcal{A}\rightarrow \mathcal{B}(\mathcal{X})$ be the bi-additive map $\phi(A,B)=\delta(A)B+A\delta(B)$, where $\delta:\mathcal{A}\rightarrow \mathcal{B}(\mathcal{X})$ is an additive map. Suppose that $\phi(A,B)=0$ for $A,B\in \mathcal{A}$ with $AB=0$. Under several conditions on $\mathcal{A}$ and $\mathcal{N}$, we show that there is an additive map $\gamma:\mathcal{A}\rightarrow \mathcal{B}(\mathcal{X})$ such that $\phi(A,B)=\gamma(AB)$ for all $A,B\in \mathcal{A}$. So there exists an additive derivation $\tau:\mathcal{A}\rightarrow \mathcal{B}(\mathcal{X})$ such that $\delta$ is equal to $\tau$ plus elementary operators. The results are a natural generalizations of several related theorems in the literature.
