Slenderness is a concept relevant to the fields of algebra, set theory, and topology. This first book on the subject is systematically presented and largely self-contained, making it ideal for ...researchers and graduate students. The appendix gives an introduction to the necessary set theory, in particular to the (non-)measurable cardinals, to help the reader make smooth progress through the text. A detailed index shows the numerous connections among the topics treated. Every chapter has a historical section to show the original sources for results and the subsequent development of ideas, and is rounded off with numerous exercises. More than 100 open problems and projects are presented, ready to inspire the keen graduate student or researcher. Many of the results are appearing in print for the first time, and many of the older results are presented in a new light.
Slenderness Dimitric, Radoslav
12/2018, Letnik:
v.Series Number 215
eBook
Slenderness is a concept relevant to the fields of algebra, set theory, and topology. This first book on the subject is systematically presented and largely self-contained, making it ideal for ...researchers and graduate students. The appendix gives an introduction to the necessary set theory, in particular to the (non-)measurable cardinals, to help the reader make smooth progress through the text. A detailed index shows the numerous connections among the topics treated. Every chapter has a historical section to show the original sources for results and the subsequent development of ideas, and is rounded off with numerous exercises. More than 100 open problems and projects are presented, ready to inspire the keen graduate student or researcher. Many of the results are appearing in print for the first time, and many of the older results are presented in a new light.
For an arbitrary infinite cardinal κ, we define classes of κ-cslender and κ-tslender modules as well as related classes of κ-hmodules and initiate a study of these classes.
Torsion and Ortho-Slender Classes Dimitric, Radoslav M
Acta applicandae mathematicae,
2008/1, Letnik:
100, Številka:
2
Journal Article
Recenzirano
This paper generalizes a number of results obtained by Dimitrić in (Glas. Mat. 21(41):327–329,
1986
; Proceedings of Hobart Conference on Rings, Modules and Radicals 1987, 204:41–50, Gordon and ...Breach,
1989
) and Dimitrić and Goldsmith in (Glas. Mat. 23(43):241–246,
1988
). The original papers were restricted to the category of Abelian groups and orthogonality was to the group of integers ℤ. Here, we are in a general Abelian category with products and coproducts, with applications to module categories and further to modules over PID’s. Another generalization is in replacing ℤ by an entire class of subobjects
of the underlying category. We examine properties of the torsion class
, Hom(
T
,
C
)=0} in relation to purity, direct summands and indecomposability as well as commutation with direct products, for example. Of special interest are members of this class when
is a class of slender objects in the ground category; in this case, members of
are called ortho-slender objects. In a sense, ortho-slenderness represents complementary, if not dual, notion to slenderness.
For an arbitrary infinite cardinal \(\kappa\), we define classes of coordinatewise \(\kappa\)-slender and tailwise \(\kappa\)-slender modules as well as related classes of \(h\kappa\)-modules and ...initiate a study of these classes.
This paper establishes grounds for deeper exploration into the question of dual nature of mathematics as an abstract discipline and as a concrete science. It is argued, as one of the consequences of ...the discussion, that the division into "pure" and "applied" mathematics is artificial. The criterion of creativity and applicability outside of the original context is used as a litmus test. It is emphasized that great societies and cultural environments produce great mathematics and individual mathematicians.
A number of examples of slender and non-slender objects are examined. Among them are polynomial rings, Noetherian rings, and Dedekind rings, as well as modules over these rings.
Objects of Type Π Dimitric, Radoslav
Slenderness,
12/2018
Book Chapter
Infinite product over infinite coproduct is an object of prime importance for slenderness. So is pure injectivity and purity. This chapter also looks into filtered quotients in general in order to ...understand the bigger picture. Countability conditions represent a watershed case. Special cases of modules and domains are examined.
Inverse Limits Dimitric, Radoslav
Slenderness,
12/2018, Letnik:
Series Number 215
Book Chapter
Properties of various inverse systems are investigated, as well as their inverse limits. The Mittag-Leffler Condition, surjectivity and flabby conditions are among the properties examined. ...Countability conditions, completions, and metrizability are intricately connected with slender objects.
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