Projective and injective modules 29 finally suppose that am. Relative projective and injective modules 2405 modules and nd injective modules, we prove that nd nd is a completecotorsion theory, where nd resp. The above results for projective modules all have duals for injective modules. Using the notion of cyclically pure injective modules, l. Some propert ies of mcpseudo injective modules are studied.
In this paper, we give some new characterizations of m,n injective. I think you will have more luck learning nontrivial facts about and examples of injective modules in noncommutative algebra texts rather than homological algebra texts. Injectiveflat modules 3 now, we explain that the inclusions can b e proper with resp ect to. Let m be a quasiprincipally injective module which a. A left rmodule m is called ncopure injective if ext1n,m0for any n. The goal is to prove that for each r module m, there is an embedding f. Pure injective modules over a commutative valuation domain. For left rmodules mand n, there is a function smodule frm,n that enjoys properties just like modules of homomorphisms in algebra. 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. M iscalled strongly sinjective, if m n for all right r modules n. Regular endomorphism rings and principally injective modules.
The category of modules over a commutative ring and abelian categories 3 f is mono,fis injective. M if and only if it is strongly simple injective in. For every left r module m, there is an injective morphism m. For every left rmodule m, there is an injective morphism m. Sam evens these notes are supplementary to class discussion, and much of them is in ash, 10. Again, the penultimate equality is using the fact that m is an r module. On m,ninjective modules and m,ncoherent rings algebra. Since every module is a factor of a free module, we have trivially the following important 3. It is well known that the class of fpprojective right rmodules, the class of fpinjective right rmodules is a complete. Maybe the most important injective module is the abelian group qz.
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. Projective and injective modules arise quite abundantly in nature. Rnand fr m,n are r modules, and in this case mr and dr. Hom rn,m 230 lim minjective modules and \prime m ideals. Various basic properties of these modules are proved, and some wellstudied rings are characterized using simpledirectinjective modules. In mathematics, especially in the area of abstract algebra known as module theory and in model theory, algebraically compact modules, also called pure injective modules, are modules that have a certain nice property which allows the solution of infinite systems of equations in the module by finitary means. Projectiveinjective modules, serre functors and symmetric. We close the paper by proving that, given a matlis domain r and an rmodule. Pdf kernelinjective module the ministry of education. If no such n exists, we set pid m modules of p injective dimension 0 are called p injective.
Strongly sinjective modules definition 1 a right rmodule m is called sninjective if every rhomomorphism f k m. Pdf we introduce and investigate ssinjectivity as a generalization of both. The linear maps coming out of 0 or going to 0 are unique, so there is no need to label them. Then there is f 2ax, monic with all other coe cients in a, such. We assume throughout the paper that ris an associative ring with identity, and that rm is a xed left rmodule. Proving that surjective endomorphisms of noetherian.
Rnof a right rmodule mand a left rmodule n, which is an smodule. A module is said to be copure injective if it is injective with respect to all modules a. If ris a ring with identity then an rmodule p is projective i one of the following equivalent conditions holds. Then we characterize commutative rings with finite self injective dimension in terms of copure injective modules. P1, homrm, n is p1injective for each p1injective module. M if and only if it is strongly simpleinjective in.
We assume throughout the paper that ris an associative ring with identity, and that rm is a xed left r module. Homomorphic images of injective modules the quarterly. The solutions to these systems allow the extension of certain kinds of module. Jir\asko introduced the concepts of l injective module as a generalization of injective module. In this paper, our aim is to present some criterions of cyclically pure injective. M i direct limit of modules m i 197 ank annihilator of a submodule k. In homological algebra texts, injective modules and more generally injective objects in an abelian category play an important technical role but are not really studied for their own s. For two fixed positive integers m and n, a right r module m is called m,n injective in case every right rhomomorphism from an ngenerated submodule of r m to m extends to one from r m to m. A module m is locally noetherian if and only if every strongly simple injective module in. Kernel injective module the ministry of education directorate general for education in wasit. We then examine the relationship between these new and known concepts. Mas an r module, as the m irun over all elements of m. The class of such modules is enveloping and falls properly between injectives and pure injectives.
The problem with this construction is that it is not. A module m over a ring is called simpledirectinjective if, whenever a and b are simple submodules of m with a. On m,n copure injective modules and m,n copure flat modules. For r r a ring, let r r mod be the category of r r modules. In our approach, projective injective modules, that is modules which are both, projec. Hyperrigid operator systems and hilbert modules shankar, p. R is said to be left m,ncoherent if each ngenerated submodule of the left rmodule r m is finitely presented. A short exact sequence of r modules is a sequence of r modules and rlinear maps 1.
Every injective r module is a direct sum of indecomposable injective r modules. M can be written as a quotient of the projective r module l r, where we have taken one copy of r for each generator of m. An r module mis divisible if for every r2rf 0gand for every m2mthere is n2msuch that rn m. Recently, some authors studied some properties of rings and modules via regularity and. The following result consider the inverse part in the case of coherent rings.
For each n, there exists a proper left injective resolution. Notes on homological algebra university of washington. Since z is a pid injective z modules are divisible z modules i. The aim of this paper is to study l injective modules and some related concepts. For a left rmodule m, we identify certain submodules of m that play a role analogous to that of prime ideals in the ring r. Projective and injective model structure on ch r shlomi agmon thursda,y july 2nd, 2012. R m, which is an injective module containing m, and has the property that any injective module containing m contains an isomorphic copy of e r m. Notes on homological algebra angus chung june 4, 2015 abstract. A module m is locally noetherian if and only if every strongly simpleinjective module in. Some results on gorenstein at modules and gorenstein nat modules are generalized. A module is called a uniform module if every two nonzero submodules have nonzero intersection.
Modules finite generation and ideals m is nitely generated as module,m quotient of ar. Pdf icpseudoinjective modules samer m saeed academia. Since product of injective modules are injective, m embeds into some injective r module, proving our theorem. Melkersson 7 provided some characterizations for a. R is said to be left m,ncoherent if each ngenerated submodule of the left r module r m is finitely presented. Let r be left artinian and let the injective envelope of every simple left rmodule be. Given modules m and n and a surjection from m to n we have the commutative diagram an.
For left r modules mand n, there is a function s module fr m,n that enjoys properties just like modules of homomorphisms in algebra. We first characterize submodules that have the extension property with respect to copure injective modules. Let m be a quasiprincipally injective module which a self. Every injective rmodule is a direct sum of indecomposable injective r modules.
It is an injective cogenerator in the category of abelian groups, which means that it is injective and any other module is contained in a suitably large product of copies of qz. Homological algebra these are notes for our rst unit on the algebraic side of homological algebra. Rnof a right r module mand a left r module n, which is an s module. Every indecomposable injective module has a local endomorphism ring. A module is called coneat injective if it is injective with respect to all coneat exact sequences. In particular we list the properties of semisimple modules, injective modules, rings of quotients, noetherian rings and dedekind domains. It is reasonable to expect that the injective modules over a principal ideal domain also have a simple structure. For a left r module m, i will identify certain submodules of m that play a role analogous to that of prime ideals. An r module m is injective provided that for every rmonomorphism g from r modules a to b,anyrhomomorphism f from a to m can be extended to an rhomomorphism h. Rmodule x is fpinjective if and only if for every rmodule m and submodule. In this paper, using geometrical invariants, we classify pure injective modules n m over a cvd v.
Also we prove that arbitrary direct sums of minjective modules are minjec tive if and only if m is locally noetherian, in the sense that every cyclic submodule of m. Every injective submodule of an injective module is a direct summand, so it is important to understand indecomposable injective modules, lam 1999, 3f. We call the module a the local injective hull of r at m, and its endomorphism ring a m end er m r end er b thelocal endomorphism ring of r at m. Generalizations of coneat injectivity, like relative coneat injectivity and full invariance of a module in its coneat injective envelope, are studied. The module m is pureinjective if any pure injective homomorphism j. The module rx is called minjective if each rhomomorphism f. Let r r be a commutative ring and c r mod c r mod the category of r r modules. A nonzero injective module is indecomposable if it is not the direct sum of nonzero injective modules. It is easy to see that injective modules are miinjective and that at modules are miat.
If ris a pid then an r module jis injective i jis divisible. For two fixed positive integers m and n, a right rmodule m is called m,n injective in case every right rhomomorphism from an ngenerated submodule of r m to m extends to one from r m to m. It turns out that a module is algebraically compact if and only if it is pureinjective. An injective module over r r is an injective object in r mod r mod. Homomorphic images of injective modules, the quarterly journal of mathematics, volume 22, issue 1, 1 march 1971, pages 8590. In this article we consider injective modules relative to a torsion theory we introduce. Then m is called rinjective if for any monomorphism f. Rnand frm,n are rmodules, and in this case mr and dr. Xde ned on a submodule kof mcan be extended to an rhomomorphism fb. In this case, the key fact is the characterization of injective z modules as divisible groups. A brief synopsis of the material included in chapter 0, we collect together basic theorems and definitions which we utilize in later chapters. In homological algebra texts, injective modules and more generally injective objects in an abelian category play an important technical role but are not really studied for their own sake.
In section three we introduce the concepts of l m injective module and sl m injective module as generalizations of m injective modules and give some results about them. All modules with finitely many elements are algebraically compact. The baer criterion and embeddings in injective modules proposition 1. With this, we can easily construct an injective resolution for every r module m. Generalizations of injective modules esperanza sanchez. R is called a right m, ncoherent ring if each ngenerated submodule of the right rmodule r m is a finitely presented right rmodule. R is called a right m, n injective ring if every right rhomomorphism from an ngenerated submodule of the right rmodule r m to r extends to one from r m to r.
Notes on tor and ext 5 which is an isomorphism if l and l. Let m be an r module, and choose a set of generators. Let xbe the submodule of v n mgenerated by the monomial tensors, q v n m xthe quotient, and q. While this is the last topic chap xx in the book, it makes. In this article, we investigate the notion of socinjectivity. In this thesis, we study the theory of projective and injective modules.
A right r module m is called quasiprincipally injective if every homomorphism from an m cyclic submodule of m to m can be extended to m. Proving that surjective endomorphisms of noetherian modules are isomorphisms and a semisimple and noetherian module is artinian. The proof proceeds by rst treating the case when r z. Definition of the ext functor suppose that m is an arbitrary r module and that the following is a projective resolution of m c 2 c 1 c 0 m 0. Similarly, the group of all rational numbers and any vector space over any eld are examples of injective modules. The module rx is called m injective if each rhomomorphism f. So in particular, every abelian group is a subgroup of an injective one. Pure injective modules relative to torsion theories. For example, all free modules that we know of, are projective modules. Pure injective modules relative to torsion theories 189 quences 0. The main reasons that i am choosing this particular topic in noncommutative algebra is for the study of representations of nite groups which we will do after the break.