Zero-dimensionality in commutative rings

Abstract:

If ${\left \{ {{R_\alpha }} \right \}_{\alpha \in A}}$ 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 $[unk]\;T$ (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.