Three theorems on form rings

Abstract:

Three theorems concerning the form ring (= associated graded ring) ${\mathbf {F}}\left ( {R,I} \right )$ of an ideal $I$ in a Noetherian ring $R$ are proved. The first characterizes, for a $P$-primary ideal in a locally quasi-unmixed ring, when ${\mathbf {F}}{\left ( {R,I} \right )_{\operatorname {red}}}$ is an integral domain in terms of when ${\mathbf {F}}{\left ( {{R_P},I{R_P}} \right )_{\operatorname {red}}}$ is an integral domain. For an aribtrary Noetherian ring $R$ the second gives a somewhat similar characterization for ${\mathbf {F}}\left ( {R,J} \right )$ to have only one prime divisor of zero for some ideal $J$ that is projectively equivalent to $I$. And the third characterizes unmixed semilocal rings in terms of the existence of an open ideal $I$ such that the zero ideal in ${\mathbf {F}}\left ( {R,I} \right )$ is isobathy.