Bulletin of the
Korean Mathematical Society BKMS

In this note we investigate some results which concern the types of local rings. In particular it is shown that if the type of a quasi-unmixed local ring $A$ is less than or equal to $\mathrm{depth} \,A + 1$, and $\hat{A}_{\p}$ is Cohen-Macaulay for every prime $\p \not= \hat{\m}$, then $A$ is Cohen-Macaulay. (This implies the previously known result: if $\hat{A}$ satisfies $(S_{n-1})$, where $n$ is the type of a ring $A$, then $A$ is Cohen-Macaulay.)

Keywords: Cohen-Macaulay ring, type of a ring, Gorenstein ring, Serre's condition

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