The classical left regular left quotient ring of a ring and its semisimplicity criteria

Abstract

Let [Formula: see text] be a ring, [Formula: see text] and [Formula: see text] be the set of regular and left regular elements of [Formula: see text] ([Formula: see text]). Goldie’s Theorem is a semisimplicity criterion for the classical left quotient ring [Formula: see text]. Semisimplicity criteria are given for the classical left regular left quotient ring [Formula: see text]. As a corollary, two new semisimplicity criteria for [Formula: see text] are obtained (in the spirit of Goldie).