Small Cofinite Irreducibles Original Research Article

We study approximately Gorenstein rings and modules which have small cofinite irreducibles (SCI). We give a criterion for a (not necessarily finitely generated) module over a local ring to have SCI via its Matlis dual. This is among other things applied to obtain Hochsterʹs characterization of finitely generated modules over a complete local ring with SCI. This is done without using anything from the theory of excellent rings. Also the relation between cyclic purity and ordinary purity is investigated for modules. Finally we study how approximately Gorenstein rings behave with respect to flat homomorphisms and with respect to regular sequences. Essential use is made of certain flat modules related to infinite sequences of ring elements.