Preprojective Modules and Auslander-Reiten Components
نویسنده
چکیده
In [2], Auslander and Smalø introduced and studied extensively preprojective modules and preinjective modules over an artin algebra. We now call a module hereditarily preprojective or hereditarily preinjective if its submodules are all preprojective or its quotient modules are all preinjective, respectively. In [4], Coelho studied Auslander-Reiten components containing only hereditarily preprojective modules and gave a number of characterizations of such components. We shall study further these modules by using the description of shapes of semi-stable Auslander-Reiten components; see [6, 7]. Our results will imply the result of Coelho [4, (1.2)] and that of Auslander-Smalø [2, (9.16)]. As an application, moreover, we shall show that a stable Auslander-Reiten component with “few” stable maps in TrD-direction is of shape ZZA∞.
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