Let $k$ be an algebraically closed field, and let
$A$ be a finite-dimensional $k$-algebra.
The first aim of classsical representation theory
was to classify all indecomposable $A$-modules,
but the homomorphisms between modules were
not considered.
One big step forward, made around 1975, was the
invention of the Auslander--Reiten quiver of $A$.
The Auslander--Reiten quiver yields a description
of the category of $A$-modules modulo its infinite
radical ${\rm rad}_A^\omega$.
But it contains no information about the
homomorphisms in ${\rm rad}_A^\omega$. We denote by mod-$A$ the category of
finite-dimensional right modules over $A$.
Let ${\rm rad}_A$ be the radical of mod-$A$.
Following Prest, we define ${\rm rad}_A^\alpha$
for each ordinal number $\alpha$ as follows:
if $J$ is an ideal in mod-$A$ and $n \geq 1$ is a
natural number, then $J^n$ is the $n$-fold product
of the ideal $J$.
For a limit number $\beta$ define
${\rm rad}_A^\beta
= \bigcap_{\alpha < \beta} {\rm rad}_A^\alpha$.
Finally, let $\alpha$ be an arbitrary infinite
ordinal number. Thus $\alpha = \beta + n$ for
some limit number $\beta$ and some natural number
$n \geq 0$.
In this case, define
${\rm rad}_A^\alpha = ({\rm rad}_A^\beta)^{n+1}$.
We denote the first limit number by $\omega$,
the second by $\omega 2$, and so on.
The minimal limit number, which is not of the form
$\omega n$ for some natural number $n \geq 1$, is
denoted by $\omega^2$. {\sc Theorem 1.} {\it
For each ordinal number $\alpha$ with
$\omega < \alpha < \omega^2$
there exists a finite-dimensional $k$-algebra $A$
with ${\rm rad}_A^\alpha \not= 0$ and
${\rm rad}_A^{\alpha + 1} = 0$. Let $Q$ be a quiver, and let $I$ be an admissible
ideal of the path algebra $kQ$.
We call $kQ/I$ {\it special biserial} if
the following hold. Any vertex of $Q$ is the starting point of
at most two arrows and also the end point of at
most two arrows. Given an arrow $\beta$, there is at most
one arrow $\alpha$ with $e(\alpha) = s(\beta)$
and $\alpha \beta \notin I$ and at most one arrow
$\gamma$ with $e(\beta) = s(\gamma)$ and
$\beta \gamma \notin I$. {\sc Theorem 2.} {\it
Let $A$ be a special biserial algebra. Then the
following are equivalent:
\begin{enumerate}
\item[{\rm (i)}] $A$ is domestic;
\item[{\rm (ii)}] ${\rm rad}_A^{\omega^2} = 0$.
\end{enumerate}
}
1991 Mathematics Subject Classification:
16G20, 16G60, 16N40