\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
Let $\varphi: (R,\fm)\to (S,\fn)$ be a flat local homomorphism of
rings. It is proved that
\begin{itemize}
\item[(a)] If $\dim S/{\fm}S>0$, then $S$ is a filter ring if and only if $R$ and $k(\fp)\otimes
_{R_{\fp}} S_{\fq}$ are Cohen-Macaulay for all $\fq\in \Spec
(S)\backslash \{\fn\}$ and $\fp=\fq\cap R$, and $S/{\fp}S$ is
catenary and equidimensional for all minimal prime ideal $\fp$ of
$R$.
\item[(b)] If $\dim S/{\fm}S=0$, then $S$ is a filter ring if and only if $R$ is a filter ring and
$k(\fp)\otimes _{R_{\fp}} S_{\fq}$ is Cohen-Macaulay for all $\fq\in
\Spec (S)\backslash \{\fn\}$ and $\fp=\fq\cap R$, and $S/{\fp}S$ is
catenary and equidimensional for all minimal prime ideal $\fp$ of
$R$.
\end{itemize}
As an application, it is shown that for a $k$-algebra $R$ and an
algebraic field extension $K$ of $k$, if $K\otimes_kR$ is locally
equidimensional then $R$ is locally filter ring if and only if
$K\otimes_kR$ is locally filter ring.
\end{document}