An rmodule jis an injective module if jsatis es one of the equivalent conditions of proposition46. They control the serre functor in the case of a quasihereditary algebra having a double centraliser with respect to a projective injective module whose endomorphism ring. Every injective submodule of an injective module is a direct summand, so it is important to understand indecomposable injective modules, lam 1999, 3f. Projective and injective modules hw pushouts and pull.
An r module m is called ding projective if there exists an exact sequence. Similarly, the group of all rational numbers and any vector space over any eld are examples of injective modules. For r r a ring, let r r mod be the category of r r modules. These are the rings in which every module is injective. An effective way to understand the behavior of a ring r is to study the various ways in which r acts on its left and right modules. Aleftrmodule e is injective if and only if every homomorphism from a left ideal l of r into e extends to a homomorphism from r into e. A module is called a uniform module if every two nonzero submodules have nonzero intersection. We classify indecomposable non projective gorensteinprojective modules over the given monomial algebra via perfect paths. In this thesis, we study the theory of projective and injective modules. Let r r be a commutative ring and c r mod c r mod the category of r r modules. Some descriptions of an fp injective envelope of an rmodule are given. In a non semisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. The gorensteinprojective modules over a monomial algebra xiaowu chen, dawei shen, guodong zhou abstract.
A right free module f over a ring r comes with a basis e i. In mathematics, especially in the area of abstract algebra known as module theory, an injective. An injective module over r r is an injective object in r mod r mod. Relative projective modules and relative injective modules lixin mao department of basic courses, nanjing institute of technology, nanjing, china and department of mathematics, nanjing university, nanjing, china nanqing ding department of mathematics, nanjing university, nanjing, china let r be a ring, and n and d. Projective and injective modules, an introduction introduction modules, or unitary modules, can act as objects in a category, with r module homomorphisms acting as morphisms. The class of all strongly gorenstein injective r modules is denoted by sgir. Mathematics and computing by bikash debnath roll no. We assume throughout the paper that ris an associative ring with identity, and that rm is a xed left r module. Note also that every pid in and particular is a hereditary ring, so this answers your second question. Both hom functors are generally only left exact, while tensor product m. In particular, an abelian group is an injective module over the ring if and only if it is divisible.
A semisimple r module is a nite direct sum of simple modules m s 1 s n and a semisimple ring is a ring rfor which all f. Various equivalent characterizations of these modules appear below. This is often proven using an injective producing lemma as in this excerpt from lams modules and rings. On localization of injective modules sciencedirect. We also investigate some connections between gorenstein projective, injective and flat modules under change of rings. Injective modules for group algebras of locally finite groups. In this st udy we di scuss t he conc epts o f project ive. The induction functor takes injective modules to injective modules, and by frobenius reciprocity, y.
Then rsr is an exact functor from amodules to asmodules which preserves essential monomorphisms and injective modules, and, hence, minimal injective resolutions. Hakmi department of mathematics, faculty of sciences, damascus university, syria. The structure of a ring over which every pureinjective resp. Foxby duality and gorenstein injective and projective modules. Hacettepejournalofmathematicsandstatistics volume4462015,14351443 applications of ngorenstein projective and injective modules xi tang y abstract. For example, for finitely generated modules over a noetherian ring, projective resolution can be taken as resolution by free modules of finite rank, but i dont see how one can easily write down injective resolutions. Projectiveinjective modules, serre functors and symmetric algebras volodymyr mazorchuk and catharina stroppel abstract we describe serre functors for generalisations of the category o associated with a semisimple complex lie algebra. Hence a projective module is merely a projective object within its category.
For example, all free modules that we know of, are projective modules. Every free module is a projective module, but the converse fails to hold over some rings, such as dedekind rings. Box 395, pretoria 0001, republic of south africa communicated by barbara l. Why are injective modules more complicated than projective. Good tilting modules and recollements of derived module categories, ii chen, hongxing and xi, changchang, journal of the mathematical society of japan, 2019. I am submitting herewith a thesis written by ryan neil campbell entitled injective modules and divisible groups. In this paper, we discuss some properties of gorenstein projective, injective and flat modules and study some connections between gorenstein injective and gorenstein flat modules. Using tools from the theory of accessible categories, injective objects are discussed in. Let a be a left noetherian ralgebra and s a multiplicatively closed set in r. Linear algebra over a ringprojective and injective modules. Naud national researchinstitute for mathematical sciences, csir, p.
Let rbe a ring with identity and let jbe an rmodule. In our approach, projectiveinjective modules, that is modules which are both, projective and injective. Applications of ngorenstein projective and injective modules. Projective and injective modules hw pushouts and pullbacks. A r module eis called injective if for each injective homomorphism f.
A module e is called the injective hull of a module m, if e is an essential extension of m, and e is injective. In section 3, we present two characterizations of cyclically pure injective modules. Then m is projective if and only if m b is a projective as a bmodule, of course. Projectiveinjective modules, serre functors and symmetric. An r module jis an injective module if jsatis es one of the equivalent conditions of proposition46. Jiri rosicky, injectivity and accessible categories. The \only if follows from the same argument as the easy direction of.
For example, it is shown that, if mr has an fp injective envelope. Every indecomposable injective module has a local endomorphism ring. Projective and injective modules arise quite abundantly in nature. If u is a subspace of a vector space v over a division ring, then u is a direct summand of v. A module over a commutative ring is injective if and only if for every ideal.
Applications of ngorenstein projective and injective modules xi tang y abstract. Characterization of projective and injective modules. This follows from this existence of injective envelopes. Injective and projective properties of representations of quivers with n edges sangwon park abstract. On injective modules and support varieties for the small quantum group christopher drupieski southeastern lie theory conference on homological methods in representation theory may 23, 2010 christopher drupieski uga on injective modules and support varieties may 23, 2010 1 14. Fp injective dimension as both a left and right module over itself. Bbe faithfully at, and ma nite amodule anoetherian. This property follows directly from the fact that zorns lemma can be used to extend a basis of u to a basis of v. The concepts of free modules, projective modules, injective modules and the like form an important area in module theory. Recently, in 10 the authors proved that a module m is pseudo injective if and only if m is invariant under any automorphism of its injective hull. Projective and injective modules play a crucial role in the study of the cohomology of representations. An r module j is injective if for any exact sequence 0. Baers criterion is discussed in many texts, for example.
However here, we will not study derivatives or integrals, but rather the notions of onetoone and onto or injective and surjective, how to compose. On gorenstein projective, injective and flat dimensions 3 gorenstein injective and. The rest of the material is from ds95, section 7, unless mentioned otherwise. Projective injective modules, serre functors and symmetric algebras volodymyr mazorchuk and catharina stroppel abstract we describe serre functors for generalisations of the category o associated with a semisimple complex lie algebra. Here, the base ring is a ring with unity, though possibly non commutative. Lams textbook contains many examples of injective modules on surrounding pages. Two recent results relate the existence of injective modules for group algebras which are small in some sense to the structure of the group. Even though i can follow the proofs they give, id like to get rid of the characterization involving free and cofree modules the fifth one. Section 4 is devoted to some applications of our previous results. Projective and injective model structure on ch r shlomi agmon thursda,y july 2nd, 2012 the material in sections 1 and 3 is taken mostly from rotman, 7. Thus, the theory of modules can be expected to be an essential chapter in the theory of rings. Sam evens these notes are supplementary to class discussion, and much of them is in ash, 10. On ding injective, ding projective and ding flat modules and complexes gillespie, james, rocky mountain journal of mathematics, 2017. Then they proved that an nstrongly gorenstein projective module.
Ding projective and ding injective modules algebra. Free modules, projective, and injective modules springerlink. If f is a free rmodule and p f is a submodule then p need not be free even if pis a direct summand of f. They control the serre functor in the case of a quasihereditary algebra having a double centraliser with respect to a projective injective module whose endomorphism ring is a symmetric algebra. Just as the gdimension has strong similarities with the projective dimension, the. Chapter 10 functions \one of the most important concepts in all of mathematics is that of function. In the process, injective modules are replaced with fp injective i. Gorenstein injective and projective modules springerlink.
I for some indexing set i so that every element in f can be uniquely written in the form. For beginners in homological algebra, it is a fact of life that injective modules seems to be more mysterious than projective modules. If r is a ring with identity then an rmodule p is projective iff one of the following equivalent conditions holds. The injective hull of an integral domain is its field of fractions, lam 1999, example 3. Annali della scuola normale superiore di pisa, classe di scienze 3e serie, tome 23, no 4 1969, p. Preparatory material for the snowbird summer school on commutative algebra these notes are intended to give the reader an idea what injective modules are, where they show up, and, to a small extent, what one can do with them. I e i r i, where all but a finite number of the elements r i. Also, several elementary results will be presented in this section, to ease reading the remainder of the paper.
Received 15082010 accepted 05072011 abstract the object of this paper is to study the endomorphism rings of locally projective and locally injective modules. Chapter 10 functions nanyang technological university. The general notion of injective objects is in section 9. A direct product of injective modules is an injective module. On injective modules and support varieties for the small. It is well known that a countably injective module is. In this paper, we develop some basic properties of such modules. Pdf n strongly gorenstein projective, injective and. An r module m is said to be nstrongly ding flat nsdflat for short, if there exists an exact sequence of r modules. In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules that is, modules with basis vectors over a ring, by keeping some of the main properties of free modules. The notion of free fuzzy modules was introduced by muganda as an extension.
Over a commutative noetherian ring, every injective module is a direct sum of indecomposable injective modules and every indecomposable injective module is the injective hull of the residue field at a prime. It is shown that each almost maximal valuation ring r such that every indecomposable injective module is countably generated. The injective hull of an injective module is itself. In this paper, we give an alternate definition for injective lmodules and prove that a direct sum of lmodules is injective if and only if each lmodule in the sum is injective. Although the class of fp injective r modules is not enveloping a class c is enveloping if every rmodule has a cenvelope see 12, theorem 4. Free modules can also be described by a universal property, but the definition given above is. As is the case with banach spaces, if one restricts the morphisms to be the contractive, module maps, then the theory of injective hilbert a modules is somewhat simpler and is largely worked out in the work of huaxin lin 23, 24 and zhou tian xu 39. I have examined the final electronic copy of this thesis for form and content and.
Let rbe a commutative noetherian ring with an identity element. The purpose of this paper is to investigate injective and projective modules relative to a hereditary torsion theory. Here, we explore rings for which every module is pseudo injective and the property. We introduce the notion of a perfect path for a monomial algebra. In our approach, projective injective modules, that is modules which are both, projective and injective, play an important role. Journal of algebra 103, 108115 1986 on localization of injective modules c. In our approach, projective injective modules, that is modules which are both, projec. Classically, modules were used in the study of representation theory see chapter 3 in first course. Hyperrigid operator systems and hilbert modules shankar, p. If f is a free r module and p f is a submodule then p need not be free even if pis a direct summand of f.