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NPP rings, reduced rings and SNF rings. (English) Zbl 1181.16010

The authors generalize the notion of pp rings and define NPP rings and study their properties. They derive properties of left NPP rings in terms of nil-injective modules. They define the notion of an NPF ring and derive properties of NPF rings in terms of reduced rings. In Section 3, the authors derive equivalent conditions for an NPP ring to be regular. Then they study GC2 rings and derive equivalent conditions for a strongly regular ring in terms of these rings. Finally they study \(n\)-regular rings and derive their properties in terms of NPP rings. Then they introduce the notion of N-flat module and extend many results of flat modules to N-flat modules. They prove that a ring \(R\) is \(n\)-regular if and only if every (cyclic) right \(R\)-module is N-flat. They conclude their paper by introducing the notions of SNF, MELT and MERT rings and derive some properties of these rings.

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
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