Zero-dimensionality in commutative rings

Abstract

If { R α } α ∈ A {\left \{ {{R_\alpha }} \right \}_{\alpha \in A}} is a family of zero-dimensional subrings of a commutative ring T T , we show that ∩ α ∈ A R α { \cap _{\alpha \in A}}{R_\alpha } is also zero-dimensional. Thus, if R R is a subring of a zero-dimensional subring [ u n k ] T [unk]\;T (a condition that is satisfied if and only if a power of r T rT is idempotent for each r ∈ R r \in R , then there exists a unique minimal zero-dimensional subring R 0 {R^0} of T T containing R R . We investigate properties of R 0 {R^0} as an R R -algebra, and we show that R 0 {R^0} is unique, up to R R -isomorphism, only if R R itself is zero-dimensional.