 
Summary: THE CLOSEDPOINT ZARISKI TOPOLOGY
FOR IRREDUCIBLE REPRESENTATIONS
K. R. Goodearl and E. S. Letzter
Abstract. In previous work, the second author introduced a topology, for spaces of irre
ducible representations, that reduces to the classical Zariski topology over commutative rings
but provides a proper refinement in various noncommutative settings. In this paper, a con
cise and elementary description of this refined Zariski topology is presented, under certain
hypotheses, for the space of simple left modules over a ring R. Namely, if R is left noetherian
(or satisfies the ascending chain condition for semiprimitive ideals), and if R is either a count
able dimensional algebra (over a field) or a ring whose (GabrielRentschler) Krull dimension
is a countable ordinal, then each closed set of the refined Zariski topology is the union of a
finite set with a Zariski closed set. The approach requires certain auxiliary results guaran
teeing embeddings of factor rings into direct products of simple modules. Analysis of these
embeddings mimics earlier work of the first author and ZimmermannHuisgen on products of
torsion modules.
1. Introduction
One of the primary obstacles to directly generalizing commutative algebraic geometry to
noncommutative contexts is the apparent absence of a "onesided" Zariski topology that
is, a noncommutative Zariski topology sensitive to left (or right) module theory. In [4], the
second author introduced a new "Zariski like" topology, on the set Irr R of isomorphism
