We study the Auslander ring of a basic left Köthe ring \(\Lambda \) and give a characterization of basic left Köthe rings in terms of their Auslander rings. We also study the functor category Mod \(((\Lambda \) -mod \()^\textrm{op})\) and characterize basic left Köthe rings \(\Lambda \) by using functor categories Mod \(((\Lambda \) -mod \()^\textrm{op})\) . As a consequence we show that there exists a bijection between the Morita equivalence classes of left Kawada rings and the Morita equivalence classes of Auslander generalized right QF-2 rings.
