  In this paper, we present a unified approach to problems of tensor product of
quotient modules of Hilbert modules over $\mathbb{C}[z]$ and corresponding
submodules of reproducing kernel Hilbert modules over $\mathbb{C}[z_1, \ldots,
z_n]$ and the doubly commutativity property of module multiplication operators
by the coordinate functions. More precisely, for a reproducing kernel Hilbert
module $\clh$ over $\mathbb{C}[z_1, \ldots, z_n]$ of analytic functions on the
polydisc in $\mathbb{C}^n$ which satisfies certain conditions, we characterize
the quotient modules $\q$ of $\clh$ such that $\q$ is of the form $\q_1 \otimes
\cdots \otimes \q_n$, for some one variable quotient modules $\{\q_1, \ldots,
\q_n\}$. For $\clh$ the Hardy module over polydisc $H^2(\mathbb{D}^n)$, this
reduces to some recent results by Izuchi, Nakazi and Seto and the third author.
This is used to obtain a classification of co-doubly commuting submodules for a
class of reproducing kernel Hilbert modules over the unit polydisc. These
results are applied to compute the cross commutators of co-doubly commuting
submodules. This is used to give further insight into the wandering subspaces
and ranks of submodules of the Hardy module case. Our results includes the case
of weighted Bergman modules over the unit polydisc in $\mathbb{C}^n$.
