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THE GRADED VERSION OF GOLDIE'S THEOREM K. R. Goodearl and J. T. Stafford
 

Summary: THE GRADED VERSION OF GOLDIE'S THEOREM
K. R. Goodearl and J. T. Stafford
Abstract. The analogue of Goldie's Theorem for prime rings is proved for rings graded by abelian
groups, eliminating unnecessary additional hypotheses used in earlier versions.
Introduction
In recent years, rings with a group­graded structure have become increasingly important and, con­
sequently, the graded analogues of Goldie's Theorems have been widely utilized. Unfortunately, the
graded result requires an awkward extra condition: given a semiprime Goldie, Z­graded ring R, one
cannot assert that R has a graded­semisimple ring of quotients unless one makes some extra assump­
tion, typically about the existence of homogeneous regular elements (see [5, Theorem C.I.1.6], for
example), or about the nondegeneracy of products of homogeneous elements (see [4, Proposition 1.4],
for instance). The standard counter­example [5, Example C.I.1.1] is the ring R = k[x] \Phi k[y], graded
by giving x degree 1 and y degree \Gamma1. Note that this ring has no homogeneous regular elements
other than units, yet it is neither graded­semisimple nor graded­artinian.
There seems to be a misconception that Goldie's Theorem for prime rings also requires such an
extra condition (see, for example, [5, xC.I.1] or [3, p. 42]) and this is awkward in applications, as
is illustrated by [2, x6.1] and [1, x5.4]. The purpose of this note is to correct this misconception
by showing that, at least for prime rings graded by abelian groups, no such extra hypotheses are
required.
A Graded Goldie Theorem

  

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

 

Collections: Mathematics