#### Date of Award

8-1996

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

David E. Dobbs

#### Committee Members

Robert McConnel, David Anderson, George Condo

#### Abstract

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.

#### Recommended Citation

Gilbert, Michael Scott, "Extensions of Commutative Rings With Linearly Ordered Intermediate Rings. " PhD diss., University of Tennessee, 1996.

https://trace.tennessee.edu/utk_graddiss/3383