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In this paper we investigate finite rank operators in the Jacobson radical ${ℛ}_{𝒩\otimes ℳ}$ of $\mathrm{A}lg(𝒩\otimes ℳ)$, where $𝒩$, $ℳ$ are nests. Based on the concrete characterizations of rank one operators in $\mathrm{A}lg(𝒩\otimes ℳ)$ and ${ℛ}_{𝒩\otimes ℳ}$, we obtain that each finite rank operator in ${ℛ}_{𝒩\otimes ℳ}$ can be written as a finite sum of rank one operators in ${ℛ}_{𝒩\otimes ℳ}$ and the weak closure of ${ℛ}_{𝒩\otimes ℳ}$ equals $\mathrm{A}lg\left(𝒩\otimes ℳ\right)$ if and only if at least one of $𝒩$, $ℳ$ is continuous.