Zero-Dimensional Extension Rings And Subrings<P>

Robert Gilmer<P>

If $\{R_{\alpha}\}_{\alpha\in A}$ is a family of zero-dimensional
subrings of a commutative ring $T$, we show that $\cap_{\alpha\in
A}R_{\alpha}$ is also zero-dimensional.  Thus, if $R$ is a subring of
a zero-dimensional subring of $T$ (a condition that is satisfied if
and only if a power of $rT$ is idempotent for each $r\in R$), then
there exists a unique minimal zero-dimensional subring $R^0$ of $T$
containing $R$.  We investigate properties of $R^0$ as an $R$-algebra,
and we show that $R^0$ is unique, up to $R$-isomorphism, only if $R$
itself is zero-dimensional.