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- Title
On a field-theoretic invariant for extensions of commutative rings, II.
- Authors
Dobbs, David E.; Gaur, Atul; Kumar, Rahul
- Abstract
This paper is a sequel. The earlier paper introduced, for any (unital) extension of (commutative unital) rings R- T, an invariant L(T=R) defined as the supremum of the lengths of chains of intermediate fields in the extension kR(Q \ R) - kT (Q), where Q runs over the prime ideals of T. Theorem 2.5 of that earlier paper calculated L(T=R) in case R - T are (commutative integral) domains such that R - T are "adjacent rings" (that is, in case R - T is a minimal ring extension of domains). The statement of that Theorem 2.5 is incorrect for some adjacent rings R - T such that R is integrally closed in T. Counterexamples are given to the original statement of Theorem 2.5. Two corrected versions of Theorem 2.5 are stated, proved and generalized from the domain-theoretic setting to the context of extensions of arbitrary rings. These results lead naturally to discussions involving the conductor (R : T) arising from a normal pair (R; T) of rings.
- Subjects
COMMUTATIVE rings; PRIME ideals
- Publication
Palestine Journal of Mathematics, 2021, Vol 10, Issue 2, p373
- ISSN
2219-5688
- Publication type
Article