Exchange Property and the Natural Preorder Between Simple Modules over a Semiartinian Ring
by
Giuseppe Baccella
Universita' di L'Aquila

The aim of the talk is to illustrate how, given a right semiartinian ring R, every irredundant set Simp_R of representatives of simple right R-modules carries a canonical structure of an artinian poset, which is a Morita invariant. We outline some basic features of this order structure; most of them bear on the fact that R satisfies the exchange property, as we proved recently. For a wide class of right semiartinian rings, which we call nice , we establish a link between those (two-sided) ideals which are pure as left ideals and some upper subsets of Simp_R.

In case R is an Artin algebra, then R is nice if and only if it is l-hereditary and there is a strict link between the poset Simp_R and the Gabriel quiver \Gamma(R); moreover there is an anti-isomorphism from the lattice of upper subsets of Simp_R to the set of all ideals which are pure as left ideals.

Finally we explain how every artinian poset (possibly after adding a suitable maximal element if it is infinite) is order isomorphic to Simp_R for some nice right semiartinian ring R.

Date received: April 19, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cage-35.