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This is the second of two volumes of a state-of-the-art survey
article collection which originates from three commutative algebra
sessions at the 2009 Fall Southeastern American Mathematical
Society Meeting at Florida Atlantic University. The articles reach
into diverse areas of commutative algebra and build a bridge
between Noetherian and non-Noetherian commutative algebra. These
volumes present current trends in two of the most active areas of
commutative algebra: non-noetherian rings (factorization, ideal
theory, integrality), and noetherian rings (the local theory,
graded situation, and interactions with combinatorics and
geometry). This volume contains surveys on aspects of closure
operations, finiteness conditions and factorization. Closure
operations on ideals and modules are a bridge between noetherian
and nonnoetherian commutative algebra. It contains a nice guide to
closure operations by Epstein, but also contains an article on test
ideals by Schwede and Tucker and one by Enescu which discusses the
action of the Frobenius on finite dimensional vector spaces both of
which are related to tight closure. Finiteness properties of rings
and modules or the lack of them come up in all aspects of
commutative algebra. However, in the study of non-noetherian rings
it is much easier to find a ring having a finite number of prime
ideals. The editors have included papers by Boynton and
Sather-Wagstaff and by Watkins that discuss the relationship of
rings with finite Krull dimension and their finite extensions.
Finiteness properties in commutative group rings are discussed in
Glaz and Schwarz's paper. And Olberding's selection presents us
with constructions that produce rings whose integral closure in
their field of fractions is not finitely generated. The final three
papers in this volume investigate factorization in a broad sense.
The first paper by Celikbas and Eubanks-Turner discusses the
partially ordered set of prime ideals of the projective line over
the integers. The editors have also included a paper on zero
divisor graphs by Coykendall, Sather-Wagstaff, Sheppardson and
Spiroff. The final paper, by Chapman and Krause, concerns
non-unique factorization.
This book contains select papers on rings, monoids and module
theory which are presented at the 3rd International Conference on
Mathematics and Statistics (AUS-ICMS 2020) held at the American
University of Sharjah, United Arab Emirates, from 6-9 February
2020. This conference was held in honour of the work of the
distinguished algebraist Daniel D. Anderson. Many participants and
colleagues from around the world felt it appropriate to acknowledge
his broad and sweeping contributions to research in algebra by
writing an edited volume in his honor. The topics covered are,
inevitably, a cross-section of the vast expansion of modern
algebra. The book is divided into two sections-surveys and recent
research developments-with each section hopefully offering
symbiotic utility to the reader. The book contains a balanced mix
of survey papers, which will enable expert and non-expert alike to
get a good overview of developments across a range of areas of
algebra. The book is expected to be of interest to both beginning
graduate students and experienced researchers.
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