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Rings, Modules, Algebras, and Abelian Groups summarizes the
proceedings of a recent algebraic conference held at Venice
International University in Italy. Surveying the most influential
developments in the field, this reference reviews the latest
research on Abelian groups, algebras and their representations,
module and ring theory, and topological algebraic structures, and
provides more than 600 current references and 570 display equations
for further exploration of the topic. It provides stimulating
discussions from world-renowned names including Laszlo Fuchs,
Robert Gilmer, Saharon Shelah, Daniel Simson, and Richard Swan to
celebrate 40 years of study on cumulative rings. Describing
emerging theories
This book collects and coherently presents the research that has
been undertaken since the author's previous book Module Theory
(1998). In addition to some of the key results since 1995, it also
discusses the development of much of the supporting material. In
the twenty years following the publication of the Camps-Dicks
theorem, the work of Facchini, Herbera, Shamsuddin, Puninski,
Prihoda and others has established the study of serial modules and
modules with semilocal endomorphism rings as one of the promising
directions for module-theoretic research. Providing readers with
insights into the directions in which the research in this field is
moving, as well as a better understanding of how it interacts with
other research areas, the book appeals to undergraduates and
graduate students as well as researchers interested in algebra.
|
New Perspectives in Algebra, Topology and Categories - Summer School, Louvain-la-Neuve, Belgium, September 12-15, 2018 and September 11-14, 2019 (Hardcover, 1st ed. 2021)
Maria Manuel Clementino, Alberto Facchini, Marino Gran
|
R2,219
Discovery Miles 22 190
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Ships in 10 - 15 working days
|
This book provides an introduction to some key subjects in algebra
and topology. It consists of comprehensive texts of some hours
courses on the preliminaries for several advanced theories in
(categorical) algebra and topology. Often, this kind of
presentations is not so easy to find in the literature, where one
begins articles by assuming a lot of knowledge in the field. This
volume can both help young researchers to quickly get into the
subject by offering a kind of " roadmap " and also help master
students to be aware of the basics of other research directions in
these fields before deciding to specialize in one of them.
Furthermore, it can be used by established researchers who need a
particular result for their own research and do not want to go
through several research papers in order to understand a single
proof. Although the chapters can be read as " self-contained "
chapters, the authors have tried to coordinate the texts in order
to make them complementary. The seven chapters of this volume
correspond to the seven courses taught in two Summer Schools that
took place in Louvain-la-Neuve in the frame of the project Fonds
d'Appui a l'Internationalisation of the Universite catholique de
Louvain to strengthen the collaborations with the universities of
Coimbra, Padova and Poitiers, within the Coimbra Group.
Occasioned by the international conference "Rings and
Factorizations" held in February 2018 at University of Graz,
Austria, this volume represents a wide range of research trends in
the theory of commutative and non-commutative rings and their
modules, including multiplicative ideal theory, Dedekind and Krull
rings and their generalizations, rings of integer
valued-polynomials, topological aspects of ring theory,
factorization theory in rings and semigroups and direct-sum
decompositions of modules. The volume will be of interest to
researchers seeking to extend or utilize work in these areas as
well as graduate students wishing to find entryways into active
areas of current research in algebra. A novel aspect of the volume
is an emphasis on how diverse types of algebraic structures and
contexts (rings, modules, semigroups, categories) may be treated
with overlapping and reinforcing approaches.
Occasioned by the international conference "Rings and
Factorizations" held in February 2018 at University of Graz,
Austria, this volume represents a wide range of research trends in
the theory of commutative and non-commutative rings and their
modules, including multiplicative ideal theory, Dedekind and Krull
rings and their generalizations, rings of integer
valued-polynomials, topological aspects of ring theory,
factorization theory in rings and semigroups and direct-sum
decompositions of modules. The volume will be of interest to
researchers seeking to extend or utilize work in these areas as
well as graduate students wishing to find entryways into active
areas of current research in algebra. A novel aspect of the volume
is an emphasis on how diverse types of algebraic structures and
contexts (rings, modules, semigroups, categories) may be treated
with overlapping and reinforcing approaches.
On the 26th of November 1992 the organizing committee gathered
together, at Luigi Salce's invitation, for the first time. The
tradition of abelian groups and modules Italian conferences (Rome
77, Udine 85, Bressanone 90) needed to be kept up by one more
meeting. Since that first time it was clear to us that our goal was
not so easy. In fact the main intended topics of abelian groups,
modules over commutative rings and non commutative rings have
become so specialized in the last years that it looked really
ambitious to fit them into only one meeting. Anyway, since everyone
of us shared the same mathematical roots, we did want to emphasize
a common link. So we elaborated the long symposium schedule: three
days of abelian groups and three days of modules over non
commutative rings with a two days' bridge of commutative algebra in
between. Many of the most famous names in these fields took part to
the meeting. Over 140 participants, both attending and contributing
the 18 Main Lectures and 64 Communications (see list on page xv)
provided a really wide audience for an Algebra meeting. Now that
the meeting is over, we can say that our initial feeling was right.
The purpose in writing this expository monograph has been
three-fold. First, the author set out to present the solution of a
problem posed by Wolfgang Krull in 1932. He asked whether what is
now called the "Krull-Schmidt Theorem" holds for artinian modules.
A negative answer was published only in 1995 by Facchini, Herbera,
Levy and Vamos. Second, the author presents the answer to a
question posed by Warfield in 1975, namely, whether the
Krull-Schmidt-Theorem holds for serial modules. Facchini published
a negative answer in 1996. The solution to the Warfield problem
shows an interesting behavior; in fact, it is a phenomena so rare
in the history of Krull-Schmidt type theorems that its presentation
to a wider mathematical audience provides the third incentive for
this monograph. Briefly, the Krull-Schmidt-Theorem holds for some,
not all, classes of modules. When it does hold, any two
indecomposable decompositions are uniquely determined up to one
permutation. For serial modules the theorem does not hold, but any
two indecomposable decompositions are uniquely determined up to two
permutations. Apart from these issues, the book addresses various
topics in module theory and ring theory, some now considered
classical (such as Goldie dimension, semiperfect rings, Krull
dimension, rings of quotients, and their applications) and others
more specialized (such as dual Goldie dimension, semilocal
endomorphism rings, serial rings and modules, exchange property,
-pure-injective modules). Open problems conclude the work.
On the 26th of November 1992 the organizing committee gathered
together, at Luigi Salce's invitation, for the first time. The
tradition of abelian groups and modules Italian conferences (Rome
77, Udine 85, Bressanone 90) needed to be kept up by one more
meeting. Since that first time it was clear to us that our goal was
not so easy. In fact the main intended topics of abelian groups,
modules over commutative rings and non commutative rings have
become so specialized in the last years that it looked really
ambitious to fit them into only one meeting. Anyway, since everyone
of us shared the same mathematical roots, we did want to emphasize
a common link. So we elaborated the long symposium schedule: three
days of abelian groups and three days of modules over non
commutative rings with a two days' bridge of commutative algebra in
between. Many of the most famous names in these fields took part to
the meeting. Over 140 participants, both attending and contributing
the 18 Main Lectures and 64 Communications (see list on page xv)
provided a really wide audience for an Algebra meeting. Now that
the meeting is over, we can say that our initial feeling was right.
|
New Perspectives in Algebra, Topology and Categories - Summer School, Louvain-la-Neuve, Belgium, September 12-15, 2018 and September 11-14, 2019 (Paperback, 1st ed. 2021)
Maria Manuel Clementino, Alberto Facchini, Marino Gran
|
R1,550
Discovery Miles 15 500
|
Ships in 10 - 15 working days
|
This book provides an introduction to some key subjects in algebra
and topology. It consists of comprehensive texts of some hours
courses on the preliminaries for several advanced theories in
(categorical) algebra and topology. Often, this kind of
presentations is not so easy to find in the literature, where one
begins articles by assuming a lot of knowledge in the field. This
volume can both help young researchers to quickly get into the
subject by offering a kind of " roadmap " and also help master
students to be aware of the basics of other research directions in
these fields before deciding to specialize in one of them.
Furthermore, it can be used by established researchers who need a
particular result for their own research and do not want to go
through several research papers in order to understand a single
proof. Although the chapters can be read as " self-contained "
chapters, the authors have tried to coordinate the texts in order
to make them complementary. The seven chapters of this volume
correspond to the seven courses taught in two Summer Schools that
took place in Louvain-la-Neuve in the frame of the project Fonds
d'Appui a l'Internationalisation of the Universite catholique de
Louvain to strengthen the collaborations with the universities of
Coimbra, Padova and Poitiers, within the Coimbra Group.
This book collects and coherently presents the research that has
been undertaken since the author's previous book Module Theory
(1998). In addition to some of the key results since 1995, it also
discusses the development of much of the supporting material. In
the twenty years following the publication of the Camps-Dicks
theorem, the work of Facchini, Herbera, Shamsuddin, Puninski,
Prihoda and others has established the study of serial modules and
modules with semilocal endomorphism rings as one of the promising
directions for module-theoretic research. Providing readers with
insights into the directions in which the research in this field is
moving, as well as a better understanding of how it interacts with
other research areas, the book appeals to undergraduates and
graduate students as well as researchers interested in algebra.
Thisexpositorymonographwaswrittenforthreereasons.
Firstly,wewantedto present the solution to a problem posed by
Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now
call the "Krull-SchmidtTheorem" holds for - tinianmodules.
Theproblemremainedopenfor63years:itssolution,anegative answer to
Krull's question, was published only in 1995 (see [Facchini,
Herbera, Levy and Vamos]). ' Secondly, we wanted to present the
answer to a question posed by War?eld in 1975 [War?eld 75]. He
proved that every ?nitely p- sented module over a serial ring is a
direct sum of uniserial modules, and asked if such a decomposition
was unique. In other words, War?eld asked whether the
"Krull-Schmidt Theorem" holds for serial modules. The solution to
this problem, a negative answer again, appeared in [Facchini 96].
Thirdly, the - lution to War?eld's problem shows interesting
behavior, a rare phenomenon in the history of Krull-Schmidt type
theorems. Essentially, the Krull-Schmidt Theorem holds for some
classes of modules and not for others. When it does hold, any two
indecomposable decompositions are uniquely determined up to a
permutation, and when it does not hold for a class of modules, this
is proved via an example. For serial modules the Krull-Schmidt
Theorem does not hold, but any two indecomposable decompositions
are uniquely determined up to two permutations. We wanted to
present such a phenomenon to a wider ma- ematical audience.
Rings, Modules, Algebras, and Abelian Groups summarizes the
proceedings of a recent algebraic conference held at Venice
International University in Italy. Surveying the most influential
developments in the field, this reference reviews the latest
research on Abelian groups, algebras and their representations,
module and ring theory, and topological algebraic structures, and
provides more than 600 current references and 570 display equations
for further exploration of the topic. It provides stimulating
discussions from world-renowned names including Laszlo Fuchs,
Robert Gilmer, Saharon Shelah, Daniel Simson, and Richard Swan to
celebrate 40 years of study on cumulative rings.
Describing emerging theories
|
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