Nearly integral homomorphisms of commutative rings

Publisher: Cambridge University Press

E-ISSN: 1755-1633|40|1|1-12

ISSN: 0004-9727

Source: Bulletin of the Australian Mathematical Society, Vol.40, Iss.1, 1989-08, pp. : 1-12

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Abstract

A unital homomorphism f: RT of commutative rings is said to be nearly integral if the induced map R/IT/IT is integral for each ideal I of R which properly contains ker (f). This concept leads to new characterisations of integral extensions and fields. For instance, if R is not a field, then an inclusion RT is integral if and only if it is nearly integral and (R, T) is a lying-over pair. It is also proved that each overring extension of an integral domain R is nearly integral if and only if dim (R) ≤ 1 and the integral closure of R is a Prüfer domain. Related properties and examples are also studied.