Extensions of Commutative Rings With Linearly Ordered Intermediate Rings

This work concerns λ-extensions of (commutative) rings, which are defined as ring extensions R ⊆ T whose set of intermediate rings is linearly ordered by inclusion. λ-extensions from a subclass of the ∆-extensions of Gilmer-Huckaba and generalize the adjacent extensions of Ferrand-Oliver, Modica, and Dechene.

In Chapter I, we characterize λ-domains (i.e., integral domains R with quotient field K such that R ⊆ K is a λ-extension), showing that they form a subclass of quasilocal i-domains. These results parallel results of Gilmer-Huckaba for ∆-domains. We relate λ-domains to divided domains and pseudovaluation domains.

Chapter II concerns λ-extensions R ⊆ T such that T is decomposable as a ring direct product. We show that a nontrivial direct product of decomposition of such a T has only two factors and characterize λ-extensions of this form, extending the corresponding result of Ferrand-Oliver for adjacent extensions.

Chapter III treats λ-extensions K ⊆ T for K a field. The case where T has a nontrivial direct product decomposition is covered using a result from Chapter II. Substantial results are obtained if T is indecomposable but not a field, generalizing results of Ferrand-Oliver and Modica for an adjacent extension T of a field K. If T is a field, we obtain good characterizations for T either purely inseparable or Galois over K. To facilitate proofs, the notion of a μ-extension of fields in introduced and related to λ-extensions of fields. We also relate infinite-dimensional λ-extensions of fields and the J-extensions of fields studied by Gilmer-Heinzer.

Chapter IV begins by studying the conductor of a λ-extension, obtaining analogues of the results of Ferrand-Oliver and Modica for adjacent extensions. We then consider λ-extensions R ⊆ T of integral domains and show that, under certain conditions (but not in generalA), the ring T is necessarily on overring of R. We classify the λ-extension overrings for two classes of integral domains and end with a useful class of examples of λ-extensions.