  Let $M$ be a free module of rank $m$ over a commutative unital ring $R$ and
let $N$ be its free submodule. We consider the problem of when a given element
of the exterior product $\Lambda^pM$ is divisible, in a sense, over elements of
the exterior product $\Lambda^r N$, where $r\le p$. Precisely, we give
conditions under which a given $\eta\in\Lambda^pM$ can be expressed as a finite
sum of elements of $\Lambda^r N$ multiplied (via the exterior product) by
elements of $\Lambda^{p-r} M$. Necessary and sufficient conditions for such
divisibility take a simple form, provided that the submodule is embedded in $M$
with singularities having the depth larger then $p-r+1$. In the special case
where $r=rank N$ the divisibility property means that $\eta=\Omega\wedge\gamma$
where $\Omega$ is the product $\omega_1\wedge\cdots\wedge\omega_r$ of elements
of a basis of $N$ and $\gamma$ is an element of $\Lambda^{p-r}M$.
  More detailed statements of these results are then used to state criteria for
existence and uniqueness of algebraic residua when the "divisor" is defined by
elements $f_1,\dots,f_k\in R$. Special cases are multidimensional logarithmic
residua in complex analysis.
