Flat Semimodules & von Neumann Regular Semirings
Jawad Abuhlail, Rangga Ganzar Noegraha
TL;DR
This paper investigates $e$-flatness for semimodules over semirings using Abuhlail's notion of exact sequences and situates it among existing flatness concepts such as $i$-flat and $m$-flat. It proves that for subtractive semirings, having every right (or left) semimodule be $e$-flat forces von Neumann regularity, linking homological flatness to structural regularity. Conversely, von Neumann regular semirings ensure that normally generated right semimodules are $m$-flat, connecting regularity with flatness in a broad module class. The work clarifies stability properties of $e$-flatness, explores Bézout-type implications, and raises open questions about which subtractive semirings exhibit universal $e$-flatness across all semimodules.
Abstract
Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of exact sequences of semimodules and its relationships with other notions of flatness for semimodules over semirings. We also prove that a subtractive semiring over which every right (left) semimodule is e-flat is a von Neumann regular semiring.