Simple flat Leavitt path algebras are von Neumann regular |
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Authors: | Ambattu Asokan Ambily Roozbeh Hazrat |
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Institution: | 1. Department of Mathematics, Cochin University of Science and Technology, Kochi, Kerala, India;2. Centre for Research in Mathematics, Western Sydney University, Sydney, Australia;3. Centre for Research in Mathematics, Western Sydney University, Sydney, Australia |
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Abstract: | For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over 40?years ago who called such rings SF-rings (i.e. simple modules are flat). In this note we show that an SF Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group, has an aperiodic unit space. For graph groupoids, this implies that the graphs are acyclic. Combining with the Abrams–Rangaswamy Theorem, it follows that SF Leavitt path algebras are regular, answering Ramamurthi’s question in positive for the class of Leavitt path algebras. |
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Keywords: | Simple flat ring von Neumann regular ring Steinberg algebra Leavitt path algebra aperoidic |
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