If \(A\) is an Artin algebra, a connected component \({\mathcal C}\) of its Auslander-Reiten quiver \(\Gamma_ A\) is a \(\pi\)-component if all indecomposables in \({\mathcal C}\) are preprojective in the sense of M. Auslander and S. O. Samlø [see J. Algebra 66, 61-122 (1980; Zbl 0477.16013)]. The main result in the paper gives other characterizations of \(\pi\)-components. One of its interesting consequences is that a \(\pi\)-component without injectives is a preprojective component in the usual sense.
There is also a study of the Auslander-Reiten quiver of an algebra \(A\) with a \(\pi\)-component \({\mathcal C}\). Such an algebra can be viewed as a triangular matrix algebra of the form \(A'\;0\choose M\;A''\) and this allows for a natural embedding of the vertices of \(\Gamma_{A'}\) into the set of vertices of \(\Gamma_ A\). It happens that they are very closely related. There are only a finite number of points of \(\Gamma_ A\) which do not lie in \(\Gamma_{A'}\), and all of them are projectives in \({\mathcal C}\). The vertices of \(\Gamma_{A'}\) belonging to \({\mathcal C}\) correspond precisely to a union of \(\pi\)-components of \(\Gamma_{A'}\). And, finally, all components of \(\Gamma_ A\) other than \({\mathcal C}\) can be individually identified with the remaining components of \(\Gamma_{A'}\).
The article is didactically written and contains a good deal of material that should be useful for any research dealing with the structure of Auslander-Reiten quivers of Artin algebras.