Survival-Pairs of Commutative Rings Have the Lying-Over Property

Abstract

Let R ⊆ T be a unital extension of commutative rings. It is proved that (R, T) is a lying-over pair (in the sense the A ⊆ B satisfies LO for all rings R ⊆ A ⊆ B ⊆ T) if (and only if) (R, T) is a survival-pair (in the sense that 𝒫T ≠ T for all rings R ⊆ A ⊆ T and all prime ideals 𝒫 of A). As a consequence, T is integral over R if (and only if) (R[X], T[X]) is a survival-pair, where X is an indeterminate over T. It is also proved that a unital homomorphism of commutative rings satisfies LO if and only if it is universally a survival homomorphism.

Key Words:

• Survival-pairs
• Lying-over pairs

2000 Mathematics Subject Classification:

• Primary 13B24, 13B21, 13B99
• Secondary 13A15, 13G05, 13B25, 14A15

Acknowledgments

The second-named author would like to acknowledge support in part by a University of Tennessee Faculty Development Award and North Dakota State University. Dobbs thanks NDSU for the warm hospitality accorded during his visit in August–September, 2000.