We study the Auslander ring of a basic left K\"othe ring $\Lambda$ and give a characterization of basic left K\"othe rings in terms of their Auslander rings. We also study the functor category Mod$((\Lambda$-mod$)^{ \rm op})$ and characterize basic left K\"othe rings $\Lambda$ by using functor categories Mod$((\Lambda$-mod$)^{ \rm op})$. 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.
Categories: math.RT math.RA