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

#### Vladimir Bavula (Sheffield)

G.209,

Let $$R$$ be a ring, $$\mathcal{C}_R$$ and $$'\mathcal{C}_R$$ be the set of regular (i.e., non-zero-divisor) and left regular elements of $$R$$, respectively ($$\mathcal{C}_R\subseteq{'\mathcal{C}_R}$$). Goldie's Theorem (1958, 1960) is a semisimplicity criterion for the classical left quotient ring $$Q_{l,cl}(R):=\mathcal{C}_R^{-1}R$$. Semisimplicity criteria are given for the classical left regular left quotient ring $$'Q_{l,cl}(R):={'\mathcal{C}_R^{-1}}R$$. As a corollary, two new semisimplicity criteria for $$Q_{l,cl}(R)$$ are obtained (in the spirit of Goldie). Applications are given for the algebra of polynomial integro-differential operators.