Multiplicative Ideal Theory in Commutative Algebra

For over forty years, Robert Gilmer's numerous articles and books have had a tremendous impact on research in commutative algebra. It is not an exaggeration to say that most articles published today in non-Noetherian ring theory, and some in Noetherian ring theory as well, originated in a topic that Gilmer either initiated or enriched by his work. This volume, a tribute to his work, consists of twenty-four articles authored by Robert Gilmer's most prominent students and followers. These articles combine surveys of past work by Gilmer and others, recent results which have never before seen print, open problems, and extensive bibliographies. In a concluding article, Robert Gilmer points out directions for future research, highlighting the open problems in the areas he considers of importance. Robert Gilmer's article is followed by the complete list of his published works, his mathematical genealogical tree, information on the writing of his four books, and reminiscences about Robert Gilmer's contributions to the stimulating research environment in commutative algebra at Florida State in the middle 1960s. The entire collection provides an in-depth overview of the topics of research in a significant and large area of commutative algebra.

Audience

This book is intended for researchers and graduate students in the field of commutative algebra.

Contenu

Commutative rngs.- Robert Gilmer's work on semigroup rings.- Numerical semigroup algebras.- Prüfer rings.- Subrings of zero-dimensional rings.- Old problems and new questions around integer-valued polynomials and factorial sequences.- Robert Gilmer's contributions to the theory of integer-valued polynomials.- Progress on the dimension question for power series rings.- Some research on chains of prime ideals influenced by the writings of Robert Gilmer.- Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings.- An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations.- Generalized Dedekind domains.- Non-unique factorizations: a survey.- Mixed polynomial/power series rings and relatinos among their spectra.- Uppers to zero in polynomial rings.- On the dimension theory of polynomial rings over pullbacks.- Almost Dedekind domains which are not Dedekind.- Integrality properties of polynomial rings and semigroup rings.- Punctually free ideals.- Holomorphy rings of function fields.- The minimal number of generators of an invertible ideal.- About minimal morphisms.- What v-coprimality can do for you.- Some questions for further research.- Robert Gilmer's published works.- Commutative Algebra at Florida State 1963-1968.