FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS

Year 2017, , 103 - 120, 17.01.2017

Rogelio Fernandez-alonso Dolors Herbera

https://doi.org/10.24330/ieja.296155

Cited By: 2

Abstract

In this paper we study some classes of rings which have a finite
lattice of preradicals. We characterize commutative rings with this condition as
finite representation type rings, i.e., artinian principal ideal rings. In general,
it is easy to see that the lattice of preradicals of a left pure semisimple ring
is a set, but it may be infinite. In fact, for a finite dimensional path algebra
Λ over an algebraically closed field we prove that Λ-pr is finite if and only if
its quiver is a disjoint union of finite quivers of type An; hence there are path
algebras of finite representation type such that its lattice of preradicals is an
infinite set. As an example, we describe the lattice of preradicals over Λ = kQ
when Q is of type An and it has the canonical orientation

Keywords

: Preradical, finite representation type

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