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 werenot considered.One big step forward, made around 1975, was theinvention 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 anatural 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.} {\itFor 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 ofat most two arrows and also the end point of atmost 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