Strongly prime ideals and strongly zero-dimensional rings

Abstract

A prime ideal [Formula: see text] is said to be strongly prime if whenever [Formula: see text] contains an intersection of ideals, [Formula: see text] contains one of the ideals in the intersection. A commutative ring with this property for every prime ideal is called strongly zero-dimensional. Some equivalent conditions are given and it is proved that a zero-dimensional ring is strongly zero-dimensional if and only if the ring is quasi-semi-local. A ring is called strongly [Formula: see text]-regular if in each ideal [Formula: see text], there is an element [Formula: see text] such that [Formula: see text] for all [Formula: see text]. Connections between the concepts strongly zero-dimensional and strongly [Formula: see text]-regular are considered.