Abstract
We prove that the multiplication ring of a centrally closed semiprime ring R has a finite rank operator over the extended centroid C iff R contains an idempotent q such that qRq is finitely generated over C and, for each , there exist and e an idempotent of C such that xz=eq.
Highlights
The symmetric ring of quotients Qs ( R) of a semiprime ring R is probably the most comfortable ring of quotients of R
The center C of Qs ( R) is called the extended centroid of R, and the C -subring QR := RC of Qs ( R) generated by R is called the central closure of R
The goal of this paper is to give a semiprime extension of the following well-known result: “If the multiplication ring of a centrally closed prime ring R has a finite rank operator over C R contains an idempotent q such that qRq is a division algebra finitely generated over C ”
Summary
The symmetric ring of quotients Qs ( R) of a semiprime ring R is probably the most comfortable ring of quotients of R. The goal of this paper is to give a semiprime extension of the following well-known result (see for instance [3], Theorem A.9): “If the multiplication ring of a centrally closed prime ring R has a finite rank operator over C R contains an idempotent q such that qRq is a division algebra finitely generated over C ”. □ As a consequence, we have the following: Corollary 1.2 Let M be a nonzero C-submodule of R and q ∈ R such that M ⊆ Cq .
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