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

Vladimir Bavula (Sheffield)


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.

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