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

Let R be a ring, CR and ′ CR be the set of regular and left regular elements of R (CR ⊆ ′ CR). Goldie’s Theorem is a semisimplicity criterion for the classical left quotient ring Ql,cl(R) := C −1 R R. Semisimplicity criteria are given for the classical left regular left quotient ring ′Ql,cl(R) := ′ C −1 R R. As a corollary, two new semisimplicity criteria for Ql,cl(R) are obtained (in the spirit of Goldie).