Zero-Dimensional Extension Rings And Subrings

Robert Gilmer

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.