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THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS
 

Summary: THE CLOSED-POINT 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 (Gabriel-Rentschler) 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 Zimmermann-Huisgen 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 "one-sided" 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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics