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Title of Thesis

Muhammad Shabir
Institute/University/Department Details
Department of Mathematics/ Quaid-i-Azam University Islamabad
Number of Pages
Keywords (Extracted from title, table of contents and abstract of thesis)
monoids, semirings, s-system, s-homomrphisms, r-semidodules, r-homomorphisms, von neumann regular semiodules, sheafs

A ring R is called regular if for each a ‚ R, there exists an element x ‚ R such that axa = a. Regular rings were introduced by von Neumann 1936, in order to clarify certain aspects of operator algebras. Since then regular rings have been very extensively studied both for their own sake, as well as for the sake of their links with operator algebras. In this thesis, we will be concerned with this important notion and some of its generalizations, from a purely algebraic point of view, in the contexts of semigroups and semirings. We will determine new characterizations of regular, weakly regular and some of the other related classes of semigroups and semirings, using algebraic and homological techniques. We will also initiate the study of sheafs for certain classes of semigroups and semirings.

Throughout this thesis, which contains five chapters, S will denote a semigroup and S-systems are representations of 5. Moreover, R will denote a semiring and R-semimodules are non-subtractive generalizations of modules over rings. Chapter 1 is of an introductory nature which provides basic definitions and reviews some of the background material which is needed for reading the subsequent chapters. In chapter 2, we introduce P-injective and divisible S-systems. We use these notions to construct an S-divisible S-system, Q(A), for an S-system A under some conditions. We also define and characterize von Neumann regular S-systems, and deduce several new characterizations of (von Neumann) regular monoids. In this chapter, we also study weakly regular monoids, and as a generalization of these monoids, we introduce the notion of normal S-systems. We show that an arbitrary monoid S is weakly regular if and only if each S-system is normal. In chapter 3, we introduce the notion of a regular semimodule, which is analogous to the notion of (von Neumann) regular S-systems studied in chapter 2. We characterize regular semimodules in terms of certain restricted injectivity properties, and use this characterization to obtain new characterizations of regular semirings. We also examine semiring analogs of the notions of hereditary, semihereditary and PP-rings. As an application of our results in this chapter, we obtain a homological characterization of PP-semirings. We also establish a characterization theorem for projective semimodules, which is analogous to the Classical Projective Basis Theorem for projective (ring) modules. In chapter 4, we define and characterize weakly regular semirings and study some properties of their prime ideal space. In chapter 5, we construct sheafs for classes of monoids and semirings, which include regular and weakly regular monoids and semirings.

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505.98 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
48.59 KB
2 1 Fundamental Concepts 1
10.35 KB
  1.1 Basic Concepts In Semigroups 1
  1.2 S-System And S-Homomrphisms 7
  1.3 Free, Projective And Injective S-Systems 11
  1.4 Semirings: Basic Definitions And Examples 14
  1.5 R-Semidodules And R-Homomorphisms 17
  1.6 Free, Projective And Injective Semimodules 19
3 2 Characterizations Of Monodies By P-Injective And Normal S-Systems 22
191.11 KB
  2.1 P-Injective And Divisible S-Systems 23
  2.2 Characterizations Of Monoids By P-Injective S-Systems 33
  2.3 Weakly Regular Monoids And Normal S-Systems 43
4 3 Regular And Pp-Semirings 54
139.1 KB
  3.1 R-Divisble And P-Injective Semimodules And Regular Semirings 55
  3.2 Pp Semirings And Pp R-Semimodules 62
  3.3 Von Neumann Regular Semiodules 66
  3.4 Projective Basis Theorem For R-Semiodules 69
5 4 Weakly Regular Semirings And Their Prime Ideal Spaces 77
71.62 KB
  4.1 Weakly Regular Semirings 77
  4.2 Prime Spectrum Of A Weakly Regular Semiring 84
6 5 Sheafs For Classes Of Monoids And Semirings 88
120.03 KB
  5.1 Sheafs Of Regular Monoids With Zero 89
  5.2 Representations Of Weakly Regular Semirings By Sections In A Presheaf 97
7 6 References 106
50.91 KB