We investigate some results which concern the types of Noetherian local rings. In particular, we show that if r(Ap) ${le}$ depth Ap + 1 for each prime ideal p of a quasi-unmixed Noetherian local ring A, then A is Cohen-Macaulay. It is also shown that the Kawasaki conjecture holds when dim A ${le}$ depth A + 1. At the end, we deal with some analogous results for modules, which are derived from the results studied on rings.