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Nearly integral homomorphisms of commutative rings

Published online by Cambridge University Press:  17 April 2009

David E. Dobbs
David E. Dobbs
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, TN. 37996–1300United States of America.
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Abstract

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A unital homomorphism f: R → T of commutative rings is said to be nearly integral if the induced map R/I → T/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 R → T 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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 1 , August 1989 , pp. 1 - 12
Copyright
Copyright © Australian Mathematical Society 1989

References

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