A CLASS OF RINGS FOR WHICH THE LATTICE OF PRERADICALS IS NOT A SET

Year 2011, Volume: 9 Issue: 9, 38 - 60, 01.06.2011

Rogelio Fernández-alonso Silvia Gavito Henry Chimal-dzul

Abstract

In this paper we define Z-coinitial rings, where Z is an integral
domain, and prove some of their properties. In particular, we characterize
commutative noetherian domains and discrete valuation domains which are
Z-coinital. We define radical modules and radical rings, and we prove that
every countable Z-coinitial and right hereditary ring is a right radical ring.
We give some examples of rings satisfying these conditions. Finally, we prove
that the lattice of preradicals of every right radical ring is not a set.

Keywords

preradical, Z-coinitial ring, radical ring, hereditary ring, Dedekind domain