  Let $\mathbf{G}$ be either a simple linear algebraic group over an
algebraically closed field of characteristic $\ell>0$ or a quantum group at an
$\ell$-th root of unity. We define a tensor ideal of singular
$\mathbf{G}$-modules in the category $\mathrm{Rep}(\mathbf{G})$ of
finite-dimensional $\mathbf{G}$-modules and study the associated quotient
category $\mathrm{\underline{Re}p}(\mathbf{G})$, called the regular quotient.
Our main results are a 'linkage principle' and a 'translation principle' for
tensor products: Let $\mathrm{\underline{Re}p}_0(\mathbf{G})$ be the essential
image in $\mathrm{\underline{Re}p}(\mathbf{G})$ of the principal block of
$\mathrm{Rep}(\mathbf{G})$. We first show that
$\mathrm{\underline{Re}p}_0(\mathbf{G})$ is closed under tensor products in
$\mathrm{\underline{Re}p}(\mathbf{G})$. Then we prove that the monoidal
structure of $\mathrm{\underline{Re}p}(\mathbf{G})$ is governed to a large
extent by the monoidal structure of $\mathrm{\underline{Re}p}_0(\mathbf{G})$.
These results can be combined to give an external tensor product decomposition
$\mathrm{\underline{Re}p}(\mathbf{G}) \cong \mathrm{Ver}(\mathbf{G}) \boxtimes
\mathrm{\underline{Re}p}_0(\mathbf{G})$, where $\mathrm{Ver}(\mathbf{G})$
denotes the Verlinde category of $\mathbf{G}$.
