MATH 101A: ALGEBRA I PART B: RINGS and MODULES in the Unit
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Complete Objects in Categories
Complete objects in categories James Richard Andrew Gray February 22, 2021 Abstract We introduce the notions of proto-complete, complete, complete˚ and strong-complete objects in pointed categories. We show under mild condi- tions on a pointed exact protomodular category that every proto-complete (respectively complete) object is the product of an abelian proto-complete (respectively complete) object and a strong-complete object. This to- gether with the observation that the trivial group is the only abelian complete group recovers a theorem of Baer classifying complete groups. In addition we generalize several theorems about groups (subgroups) with trivial center (respectively, centralizer), and provide a categorical explana- tion behind why the derivation algebra of a perfect Lie algebra with trivial center and the automorphism group of a non-abelian (characteristically) simple group are strong-complete. 1 Introduction Recall that Carmichael [19] called a group G complete if it has trivial cen- ter and each automorphism is inner. For each group G there is a canonical homomorphism cG from G to AutpGq, the automorphism group of G. This ho- momorphism assigns to each g in G the inner automorphism which sends each x in G to gxg´1. It can be readily seen that a group G is complete if and only if cG is an isomorphism. Baer [1] showed that a group G is complete if and only if every normal monomorphism with domain G is a split monomorphism. We call an object in a pointed category complete if it satisfies this latter condi- arXiv:2102.09834v1 [math.CT] 19 Feb 2021 tion. -
On the Cohen-Macaulay Modules of Graded Subrings
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 2, Pages 735{756 S 0002-9947(04)03562-7 Article electronically published on April 27, 2004 ON THE COHEN-MACAULAY MODULES OF GRADED SUBRINGS DOUGLAS HANES Abstract. We give several characterizations for the linearity property for a maximal Cohen-Macaulay module over a local or graded ring, as well as proofs of existence in some new cases. In particular, we prove that the existence of such modules is preserved when taking Segre products, as well as when passing to Veronese subrings in low dimensions. The former result even yields new results on the existence of finitely generated maximal Cohen-Macaulay modules over non-Cohen-Macaulay rings. 1. Introduction and definitions The notion of a linear maximal Cohen-Macaulay module over a local ring (R; m) was introduced by Ulrich [17], who gave a simple characterization of the Gorenstein property for a Cohen-Macaulay local ring possessing a maximal Cohen-Macaulay module which is sufficiently close to being linear, as defined below. A maximal Cohen-Macaulay module (abbreviated MCM module)overR is one whose depth is equal to the Krull dimension of the ring R. All modules to be considered in this paper are assumed to be finitely generated. The minimal number of generators of an R-module M, which is equal to the (R=m)-vector space dimension of M=mM, will be denoted throughout by ν(M). In general, the length of a finitely generated Artinian module M is denoted by `(M)(orby`R(M), if it is necessary to specify the ring acting on M). -
Modules and Lie Semialgebras Over Semirings with a Negation Map 3
MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP GUY BLACHAR Abstract. In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. We define a notion of lifting a module with a negation map, similarly to the tropicalization process, and use it to prove several theorems about semirings with a negation map which possess a lift. In the context of Lie semialgebras over semirings with a negation map, we first give basic definitions, and provide parallel constructions to the classical Lie algebras. We prove an ELT version of Cartan’s criterion for semisimplicity, and provide a counterexample for the naive version of the PBW Theorem. Contents Page 0. Introduction 2 0.1. Semirings with a Negation Map 2 0.2. Modules Over Semirings with a Negation Map 3 0.3. Supertropical Algebras 4 0.4. Exploded Layered Tropical Algebras 4 1. Modules over Semirings with a Negation Map 5 1.1. The Surpassing Relation for Modules 6 1.2. Basic Definitions for Modules 7 1.3. -morphisms 9 1.4. Lifting a Module Over a Semiring with a Negation Map 10 1.5. Linearly Independent Sets 13 1.6. d-bases and s-bases 14 1.7. Free Modules over Semirings with a Negation Map 18 2. -
Irreducible Representations of Finite Monoids
U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately. -
Research.Pdf (1003.Kb)
PERSISTENT HOMOLOGY: CATEGORICAL STRUCTURAL THEOREM AND STABILITY THROUGH REPRESENTATIONS OF QUIVERS A Dissertation presented to the Faculty of the Graduate School, University of Missouri, Columbia In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by KILLIAN MEEHAN Dr. Calin Chindris, Dissertation Supervisor Dr. Jan Segert, Dissertation Supervisor MAY 2018 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled PERSISTENT HOMOLOGY: CATEGORICAL STRUCTURAL THEOREM AND STABILITY THROUGH REPRESENTATIONS OF QUIVERS presented by Killian Meehan, a candidate for the degree of Doctor of Philosophy of Mathematics, and hereby certify that in their opinion it is worthy of acceptance. Associate Professor Calin Chindris Associate Professor Jan Segert Assistant Professor David C. Meyer Associate Professor Mihail Popescu ACKNOWLEDGEMENTS To my thesis advisors, Calin Chindris and Jan Segert, for your guidance, humor, and candid conversations. David Meyer, for our emphatic sharing of ideas, as well as the savage question- ing of even the most minute assumptions. Working together has been an absolute blast. Jacob Clark, Brett Collins, Melissa Emory, and Andrei Pavlichenko for conver- sations both professional and ridiculous. My time here was all the more fun with your friendships and our collective absurdity. Adam Koszela and Stephen Herman for proving that the best balm for a tired mind is to spend hours discussing science, fiction, and intersection of the two. My brothers, for always reminding me that imagination and creativity are the loci of a fulfilling life. My parents, for teaching me that the best ideas are never found in an intellec- tual vacuum. My grandparents, for asking me the questions that led me to where I am. -
Pure Injective Modules Relative to Torsion Theories
International Journal of Algebra, Vol. 8, 2014, no. 4, 187 - 194 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.429 Pure Injective Modules Relative to Torsion Theories Mehdi Sadik Abbas and Mohanad Farhan Hamid Department of Mathematics, University of Mustansiriya, Baghdad, Iraq Copyright c 2014 Mehdi Sadik Abbas and Mohanad Farhan Hamid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let τ be a hereditary torsion theory on the category Mod-R of right R-modules. A right R-module M is called pure τ-injective if it is injec- tive with respect to every pure exact sequence having a τ-torsion cok- ernel. Every module has a pure τ-injective envelope. A module M is called purely quasi τ-injective if it is fully invariant in its pure τ-injective envelope. A module M is called quasi pure τ-injective if maps from τ- dense pure submodules of M into M are extendable to endomorphisms of M. The class of pure τ-injective modules is properly contained in the class of purely quasi τ-injective modules which is in turn properly contained in the class of quasi pure τ-injectives. A torsion theoretic version of each of the concepts of regular and pure semisimple rings is characterized using the above generalizations of pure injectivity. Mathematics Subject Classification: 16D50 Keywords: (quasi) pure injective module, purely quasi injective module, torsion theory 1 Introduction We will denote by R an associative ring with a nonzero identity and by τ = (T ; F) a hereditary torsion theory on the category Mod-R of right R-modules. -
Cyclic Pure Submodules
International Journal of Algebra, Vol. 3, 2009, no. 3, 125 - 135 Cyclic Pure Submodules V. A. Hiremath1 Department of Mathematics, Karnatak University Dharwad-580003, India va hiremath@rediffmail.com Seema S. Gramopadhye2 Department of Mathematics, Karnatak University Dharwad-580003, India e-mail:[email protected] Abstract P.M.Cohn [3] has introduced the notion of purity for R-modules. With respect to purity, flat, absolutely pure and regular modules are studied. In this paper we introduce and study the corresponding no- tions of c-flat, absolutely c-pure and c-regular modules for cyclic pu- rity. We prove that absolutely c-pure R-modules are precisely injective modules. Also we study the relationship between c-flat and torsion-free modules over commutative integral domains and non-commutative non- integral domains. Also, we study the conditions under which c-regular R-modules are semi-simple. Mathematics Subject Classifications: 16D40, 16D50 Keywords: Pure submodules, flat module and regular ring Introduction In this paper, by a ring R we mean an associative ring with unity and by an R-module we mean a unitary right R-module. Z(M) denotes the singular submodule of the R-module M. 1Corresponding author 2The author is supported by University Research Scholarship. 126 V. A. Hiremath and S. S. Gramopadhye A ring R is said to be principal projective, if every principal right ideal is projective. We denote this ring by p.p. The notion of purity has an important role in module theory and in model theory. In model theory, the notion of pure exact sequence is more useful than split exact sequences. -
Separable Commutative Rings in the Stable Module Category of Cyclic Groups
SEPARABLE COMMUTATIVE RINGS IN THE STABLE MODULE CATEGORY OF CYCLIC GROUPS PAUL BALMER AND JON F. CARLSON Abstract. We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group. Contents Introduction1 1. Separable ring-objects4 2. The Kelly radical and the tensor6 3. The case of the group of prime order 14 4. The case of the general cyclic group 16 References 18 Introduction Since 1960 and the work of Auslander and Goldman [AG60], an algebra A over op a commutative ring R is called separable if A is projective as an A ⊗R A -module. This notion turns out to be remarkably important in many other contexts, where the module category C = R- Mod and its tensor ⊗ = ⊗R are replaced by an arbitrary tensor category (C; ⊗). A ring-object A in such a category C is separable if multiplication µ : A⊗A ! A admits a section σ : A ! A⊗A as an A-A-bimodule in C. See details in Section1. Our main result (Theorem 4.1) concerns itself with modular representation theory of finite groups: Main Theorem. Let | be a separably closed field of characteristic p > 0 and let G be a cyclic p-group. Let A be a commutative and separable ring-object in the stable category |G- stmod of finitely generated |G-modules modulo projectives. -
Math 250A: Groups, Rings, and Fields. H. W. Lenstra Jr. 1. Prerequisites
Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites This section consists of an enumeration of terms from elementary set theory and algebra. You are supposed to be familiar with their definitions and basic properties. Set theory. Sets, subsets, the empty set , operations on sets (union, intersection, ; product), maps, composition of maps, injective maps, surjective maps, bijective maps, the identity map 1X of a set X, inverses of maps. Relations, equivalence relations, equivalence classes, partial and total orderings, the cardinality #X of a set X. The principle of math- ematical induction. Zorn's lemma will be assumed in a number of exercises. Later in the course the terminology and a few basic results from point set topology may come in useful. Group theory. Groups, multiplicative and additive notation, the unit element 1 (or the zero element 0), abelian groups, cyclic groups, the order of a group or of an element, Fermat's little theorem, products of groups, subgroups, generators for subgroups, left cosets aH, right cosets, the coset spaces G=H and H G, the index (G : H), the theorem of n Lagrange, group homomorphisms, isomorphisms, automorphisms, normal subgroups, the factor group G=N and the canonical map G G=N, homomorphism theorems, the Jordan- ! H¨older theorem (see Exercise 1.4), the commutator subgroup [G; G], the center Z(G) (see Exercise 1.12), the group Aut G of automorphisms of G, inner automorphisms. Examples of groups: the group Sym X of permutations of a set X, the symmetric group S = Sym 1; 2; : : : ; n , cycles of permutations, even and odd permutations, the alternating n f g group A , the dihedral group D = (1 2 : : : n); (1 n 1)(2 n 2) : : : , the Klein four group n n h − − i V , the quaternion group Q = 1; i; j; ij (with ii = jj = 1, ji = ij) of order 4 8 { g − − 8, additive groups of rings, the group Gl(n; R) of invertible n n-matrices over a ring R. -
Mel Hochster's Lecture Notes
MATH 614 LECTURE NOTES, FALL, 2017 by Mel Hochster Lecture of September 6 We assume familiarity with the notions of ring, ideal, module, and with the polynomial ring in one or finitely many variables over a commutative ring, as well as with homomor- phisms of rings and homomorphisms of R-modules over the ring R. As a matter of notation, N ⊆ Z ⊆ Q ⊆ R ⊆ C are the non-negative integers, the integers, the rational numbers, the real numbers, and the complex numbers, respectively, throughout this course. Unless otherwise specified, all rings are commutative, associative, and have a multiplica- tive identity 1 (when precision is needed we write 1R for the identity in the ring R). It is possible that 1 = 0, in which case the ring is f0g, since for every r 2 R, r = r ·1 = r ·0 = 0. We shall assume that a homomorphism h of rings R ! S preserves the identity, i.e., that h(1R) = 1S. We shall also assume that all given modules M over a ring R are unital, i.e., that 1R · m = m for all m 2 M. When R and S are rings we write S = R[θ1; : : : ; θn] to mean that S is generated as a ring over its subring R by the elements θ1; : : : ; θn. This means that S contains R and the elements θ1; : : : ; θn, and that no strictly smaller subring of S contains R and the θ1; : : : ; θn. It also means that every element of S can be written (not necessarily k1 kn uniquely) as an R-linear combination of the monomials θ1 ··· θn . -
Homological Algebra Lecture 1
Homological Algebra Lecture 1 Richard Crew Richard Crew Homological Algebra Lecture 1 1 / 21 Additive Categories Categories of modules over a ring have many special features that categories in general do not have. For example the Hom sets are actually abelian groups. Products and coproducts are representable, and one can form kernels and cokernels. The notation of an abelian category axiomatizes this structure. This is useful when one wants to perform module-like constructions on categories that are not module categories, but have all the requisite structure. We approach this concept in stages. A preadditive category is one in which one can add morphisms in a way compatible with the category structure. An additive category is a preadditive category in which finite coproducts are representable and have an \identity object." A preabelian category is an additive category in which kernels and cokernels exist, and finally an abelian category is one in which they behave sensibly. Richard Crew Homological Algebra Lecture 1 2 / 21 Definition A preadditive category is a category C for which each Hom set has an abelian group structure satisfying the following conditions: For all morphisms f : X ! X 0, g : Y ! Y 0 in C the maps 0 0 HomC(X ; Y ) ! HomC(X ; Y ); HomC(X ; Y ) ! HomC(X ; Y ) induced by f and g are homomorphisms. The composition maps HomC(Y ; Z) × HomC(X ; Y ) ! HomC(X ; Z)(g; f ) 7! g ◦ f are bilinear. The group law on the Hom sets will always be written additively, so the last condition means that (f + g) ◦ h = (f ◦ h) + (g ◦ h); f ◦ (g + h) = (f ◦ g) + (f ◦ h): Richard Crew Homological Algebra Lecture 1 3 / 21 We denote by 0 the identity of any Hom set, so the bilinearity of composition implies that f ◦ 0 = 0 ◦ f = 0 for any morphism f in C. -
Lifting Defects for Nonstable K 0-Theory of Exchange Rings And
LIFTING DEFECTS FOR NONSTABLE K0-THEORY OF EXCHANGE RINGS AND C*-ALGEBRAS FRIEDRICH WEHRUNG Abstract. The assignment (nonstable K0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (i) There is no functor Γ, from simplicial monoids with order-unit with nor- malized positive homomorphisms to exchange rings, such that V ◦ Γ =∼ id. (ii) There is no functor Γ, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that V ◦ Γ =∼ id. 3 (iii) There is a {0, 1} -indexed commutative diagram D~ of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0. By using categorical tools (larders, lifters, CLL) from a recent book from the author with P. Gillibert, we deduce that there exists a unital exchange ring of cardinality ℵ3 (resp., an ℵ3-separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(R) is the positive cone of a dimension group but it is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.