A Sufficient Condition for a Minimal Ring Extension to Be an Overring

Abstract

It is proved that if R ⊂ T is a minimal ring extension such that (a) the set of zero-divisors of R is contained in a nonmaximal prime ideal of R and (b) each non-zero-divisor of R remains a non-zero-divisor in T, then T is isomorphic to an R-subalgebra of the total quotient ring of R. A nondomain example R ⊂ T satisfying the above hypotheses is given where the Krull dimension of R is any preassigned integer n ≥ 2 and the total quotient ring of R has Krull dimension n − 1. Examples also show that neither of the hypothesis (a), (b) can be deleted. In the case that T is a domain, the above result recovers a theorem of Sato et al. (1992).

Key Words:

• Domain
• Krull dimension
• Minimal ring extension
• Nonmaximal prime ideal
• Overring
• Zero-divisor

2000 Mathematics Subject Classification:

• Primary 13B99, 13A15
• Secondary 13G05, 13B30, 13C15

Notes

Communicated by J. Kuzmanovich.