The envelope \(E(B)\) of a submodule \(B\subset M\) of a module \(M\) over a commutative ring \(R\) is a module version of the radical of an ideal. This paper introduces the \(n\)th envelope \(E_n(B) = E(\langle E_{n-1}(B)\rangle), E_0(B)=B\). It is shown that if \(R\) is an arithmetical ring of finite Krull dimension \(n\), then \(\langle E_n(B)\rangle = \text{rad}(B)\).