Toward homological structure theory of semimodules: On semirings all of whose cyclic semimodules are projective
TL;DR: In this paper, the homological structure theory of semirings and CP-semirings was introduced, and the properties of semimodules over Boolean algebras whose endomorphism semimings are projective were studied.
About: This article is published in Journal of Algebra.The article was published on 2017-04-15 and is currently open access. It has received 16 citations till now. The article focuses on the topics: Endomorphism.
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TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.
1,479 citations
TL;DR: In this article, the authors introduce and study e-injective semimodules, in particular over additively idempotent semirings, and give complete characterizations of bounded distributive lattices, subtractive semidempotents, and simple semidimodels.
Abstract: In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize one-sided Noetherian rings in terms of direct sums of e-injective semimodules. Also, we give complete characterizations of bounded distributive lattices, subtractive semirings, and simple semirings, all of whose cyclic (finitely generated) semimodules are e-injective.
12 citations
TL;DR: In this article, the pre-ordered abelian Grothendieck groups K 0 (S ) and S K 0(S ) of the isomorphism classes of the finitely generated projective and strongly projective S-semimodules, respectively, over an arbitrary semiring S, are shown to be complete invariants of ultramatricial algebras.
6 citations
TL;DR: In this article, it was shown that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring.
Abstract: We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.
4 citations
TL;DR: In this article, the authors give a characterization of injective semimodules over additively idempotent semirings, and give an explicit construction of the injective hulls of semi-modules.
3 citations
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Book•
01 Jan 1971
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Abstract: I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.
9,254 citations
Book•
01 Jan 1974
TL;DR: In this paper, the authors provide a self-contained account of much of the theory of rings and modules, focusing on the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules.
Abstract: This book is intended to provide a self-contained account of much of the theory of rings and modules. The theme of the text throughout is the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules. Following a brief outline of the foundations, the book begins with the basic definitions and properties of rings, modules and homomorphisms. The remainder of the text gives comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, decomposition theory, and semiperfect and perfect rings. This second edition includes a chapter containing many of the classical results on Artinian rings that have helped form the foundation for much of contemporary research on the representation theory of Artinian rings and finite-dimensional algebras.
2,650 citations
Book•
01 Jan 1999
TL;DR: In this article, Baer's Test for Injectivity is used to evaluate the suitability of a set of Injective Modules over a group of Invertible Fractional Ideals.
Abstract: 1 Free Modules, Projective, and Injective Modules.- 1. Free Modules.- 1A. Invariant Basis Number (IBN).- 1B. Stable Finiteness.- 1C. The Rank Condition.- 1D. The Strong Rank Condition.- 1E. Synopsis.- Exercises for 1.- 2. Projective Modules.- 2A. Basic Definitions and Examples.- 2B. Dual Basis Lemma and Invertible Modules.- 2C. Invertible Fractional Ideals.- 2D. The Picard Group of a Commutative Ring.- 2E. Hereditary and Semihereditary Rings.- 2F. Chase Small Examples.- 2G. Hereditary Artinian Rings.- 2H. Trace Ideals.- Exercises for 2.- 3. Injective Modules.- 3A. Baer's Test for Injectivity.- 3B. Self-Injective Rings.- 3C. Injectivity versus Divisibility.- 3D. Essential Extensions and Injective Hulls.- 3E. Injectives over Right Noetherian Rings.- 3F. Indecomposable Injectives and Uniform Modules.- 3G. Injectives over Some Artinian Rings.- 3H. Simple Injectives.- 31. Matlis' Theory.- 3J. Some Computations of Injective Hulls.- 3K. Applications to Chain Conditions.- Exercises for 3.- 2 Flat Modules and Homological Dimensions.- 4. Flat and Faithfully Flat Modules.- 4A. Basic Properties and Flatness Tests.- 4B. Flatness, Torsion-Freeness, and von Neumann Regularity.- 4C. More Flatness Tests.- 4D. Finitely Presented (f.p.) Modules.- 4E. Finitely Generated Flat Modules.- 4F. Direct Products of Flat Modules.- 4G. Coherent Modules and Coherent Rings.- 4H. Semihereditary Rings Revisited.- 41. Faithfully Flat Modules.- 4J. Pure Exact Sequences.- Exercises for 4.- 5. Homological Dimensions.- 5A. Schanuel's Lemma and Projective Dimensions.- 5B. Change of Rings.- 5C. Injective Dimensions.- 5D. Weak Dimensions of Rings.- 5E. Global Dimensions of Semiprimary Rings.- 5F. Global Dimensions of Local Rings.- 5G. Global Dimensions of Commutative Noetherian Rings.- Exercises for 5.- 3 More Theory of Modules.- 6. Uniform Dimensions, Complements, and CS Modules.- 6A. Basic Definitions and Properties.- 6B. Complements and Closed Submodules.- 6C. Exact Sequences and Essential Closures.- 6D. CS Modules: Two Applications.- 6E. Finiteness Conditions on Rings.- 6F. Change of Rings.- 6G. Quasi-Injective Modules.- Exercises for 6.- 7. Singular Submodules and Nonsingular Rings.- 7A. Basic Definitions and Examples.- 7B. Nilpotency of the Right Singular Ideal.- 7C. Goldie Closures and the Reduced Rank.- 7D. Baer Rings and Rickart Rings.- 7E. Applications to Hereditary and Semihereditary Rings.- Exercises for 7.- 8. Dense Submodules and Rational Hulls.- 8A. Basic Definitions and Examples.- 8B. Rational Hull of a Module.- 8C. Right Kasch Rings.- Exercises for 8.- 4 Rings of Quotients.- 9. Noncommutative Localization.- 9A. "The Good'.- 9B. "The Bad'.- 9C. "The Ugly".- 9D. An Embedding Theorem of A. Robinson.- Exercises for 9.- 10. Classical Rings of Quotients.- 10A. Ore Localizations.- 10B. Right Ore Rings and Domains.- 10C. Polynomial Rings and Power Series Rings.- 10D. Extensions and Contractions.- Exercises for 10.- 11. Right Goldie Rings and Goldie's Theorems.- 11A. Examples of Right Orders.- 11B. Right Orders in Semisimple Rings.- 11C. Some Applications of Goldie's Theorems.- 11D. Semiprime Rings.- 11E. Nil Multiplicatively Closed Sets.- Exercises for 11.- 12. Artinian Rings of Quotients.- 12A. Goldie's ?-Rank.- 12B. Right Orders in Right Artinian Rings.- 12C. The Commutative Case.- 12D. Noetherian Rings Need Not Be Ore.- Exercises for 12.- 5 More Rings of Quotients.- 13. Maximal Rings of Quotients.- 13A. Endomorphism Ring of a Quasi-Injective Module.- 13B. Construction of Qrmax(R).- 13C. Another Description of Qrmax(R).- 13D. Theorems of Johnson and Gabriel.- Exercises for 13.- 14. Martindale Rings of Quotients.- 14A. Semiprime Rings Revisited.- 14B. The Rings Qr(R) and Qs(R).- 14C. The Extended Centroid.- 14D. Characterizations of and Qr(R) and Qs(R).- 14E. X-Inner Automorphisms.- 14F. A Matrix Ring Example.- Exercises for 14.- 6 Frobenius and Quasi-Frobenius Rings.- 15. Quasi-Frobenius Rings.- 15A. Basic Definitions of QF Rings.- 15B. Projectives and Injectives.- 15C. Duality Properties.- 15D. Commutative QF Rings, and Examples.- Exercises for 15.- 16. Frobenius Rings and Symmetric Algebras.- 16A. The Nakayama Permutation.- 16B. Definition of a Frobenius Ring.- 16C. Frobenius Algebras and QF Algebras.- 16D. Dimension Characterizations of Frobenius Algebras.- 16E. The Nakayama Automorphism.- 16F. Symmetric Algebras.- 16G. Why Frobenius?.- Exercises for 16.- 7 Matrix Rings, Categories of Modules, and Morita Theory.- 17. Matrix Rings.- 17A. Characterizations and Examples.- 17B. First Instance of Module Category Equivalences.- 17C. Uniqueness of the Coefficient Ring.- Exercises for 17.- 18. Morita Theory of Category Equivalences.- 18A. Categorical Properties.- 18B. Generators and Progenerators.- 18C. The Morita Context.- 18D. Morita I, II, III.- 18E. Consequences of the Morita Theorems.- 18F. The Category ? [M].- Exercises for 18.- 19. Morita Duality Theory.- 19A. Finite Cogeneration and Cogenerators.- 19B. Cogenerator Rings.- 19C. Classical Examples of Dualities.- 19D. Morita Dualities: Morita I.- 19E. Consequences of Morita I.- 19F. Linear Compactness and Reflexivity.- 19G. Morita Dualities: Morita II.- Exercises for 19.- References.- Name Index.
1,584 citations
Book•
09 Jan 2018
TL;DR: Topos theory has been studied at the graduate student level for a long time, see as discussed by the authors for an overview of the main applications of topos in algebraic geometry and logic.
Abstract: This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds (algebraic, analytic, etc.). Sheaves also appear in logic as carriers for models of set theory as well as for the semantics of other types of logic. Grothendieck introduced a topos as a category of sheaves for algebraic geometry. Subsequently, Lawvere and Tierney obtained elementary axioms for such (more general) categories. This introduction to topos theory begins with a number of illustrative examples that explain the origin of these ideas and then describes the sheafification process and the properties of an elementary topos. The applications to axiomatic set theory and the use in forcing (the Independence of the Continuum Hypothesis and of the Axiom of Choice) are then described. Geometric morphisms- like continuous maps of spaces and the construction of classifying topoi, for example those related to local rings and simplicial sets, next appear, followed by the use of locales (pointless spaces) and the construction of topoi related to geometric languages and logic. This is the first text to address all of these varied aspects of topos theory at the graduate student level.
1,166 citations
Book•
01 Jan 1991
TL;DR: In this article, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.
1,090 citations