Journal of the
Korean Mathematical Society JKMS

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

Keywords: Cohen-Macaulay ring, type of a ring, Gorenstein ring

MSC numbers: 13C14, 13C15, 13D07, 13D45, 13H10