The classical left regular left quotient ring of a ring and its semisimplicity criteria. (English) Zbl 1367.16027
Summary: Let \(R\) be a ring, \(\mathcal{C}_R\) and \(^\prime \mathcal{C}_R\) be the set of regular and left regular elements of \(R(\mathcal{C}_R\subseteq {}^\prime \mathcal{C}_R)\). Goldie’s theorem is a semisimplicity criterion for the classical left quotient ring \(Q_{l,cl}(R):= {}^\prime \mathcal{C}_R^{-1}R\). Semisimplicity criteria are given for the classical left regular left quotient ring \({}^\prime Q_{l,cl}(R):= {}^\prime \mathcal{C}_R^{-1}R\). As a corollary, two new semisimplicity criteria for \(Q_{l,cl}(R)\) are obtained (in the spirit of Goldie).
MSC:
| 16S85 | Associative rings of fractions and localizations |
| 16P20 | Artinian rings and modules (associative rings and algebras) |
| 16U20 | Ore rings, multiplicative sets, Ore localization |
Keywords:
Goldie’s theorem; (classical) left quotient ring; (classical) left regular left quotient ringReferences:
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