For example, the injective hull of uaannuas is isomorphic to the direct sum of d copies of the injective hull of any simple a module s of dimension d1 cf. In section three we introduce the concepts of lm injective module and slm injective module as generalizations of m injective modules and give some results about them. A function, f, is called injective if it is onetoone. The point is that wecanextenda qtorif weextendqtothelargermoduler.
A module is called a uniform module if every two nonzero submodules have nonzero intersection. If ris a pid then an r module jis injective i jis divisible. They are important tools in generalizing results on commutative algebra to noncommutative case and a starting point in this direction was the work of matlis on injective modules over noetherian rings 44. In this thesis, we study the theory of projective and injective modules. Strongly s injective modules definition 1 a right r module m is called s n injective if every r homomorphism f k m.
Rmodule is always an isomorphism if and only if r is odimensional, that is, if every prime ideal is maximal. Let p be a projective r module and let i be an ideal of r and. Some examples on provingdisproving a function is injective surjective csci 2824, spring 2015 this page contains some examples that should help you finish assignment 6. In this section, we define these concepts officially in terms of preimages, and explore some easy examples and consequences.
This is often proven using an injective producing lemma as in this excerpt from lams modules and rings. A function an injective onetoone function a surjective onto function a bijective onetoone and onto function a few words about notation. Using the concepts of general principally injective modules, we will study some characterizations of regular modules and their endomorphism ring. In section 3, we present two characterizations of cyclically pure injective modules. Surjective functions are matchmakers who make sure they find a match for all of set b, and who dont mind using polyamory to do it. Functions may be surjective or onto there are also surjective functions. Informally, an injection has each output mapped to by at most one input, a surjection includes the entire possible range in the output, and a bijection has both conditions be true. Regular endomorphism rings and principally injective modules. It is called bijective if it is both onetoone and onto. The category of modules over a commutative ring and abelian categories 3 f is mono,fis injective. Injective modules, introduced by baer, eckmann, and schopf, are the building blocks in the theory of noetherian rings. Similarly, the group of all rational numbers and any vector space over any eld are examples of injective modules.
Relative projective and injective modules 2405 modules and nd injective modules, we prove that nd nd is a completecotorsion theory, where nd resp. 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. A bof r modules, the associated homomorphism of abelian groups f hom rb,e hom. We prove that weakly injective bck modules are characterized by baers criterion, which we use to prove that over principal bounded implicative bcks, every module is weakly injective corollary 3. Let a be any nonnoetherian ring, let b be the field of 2 elements, and let c be the field of rational numbers. Math 3000 injective, surjective, and bijective functions. German football players dressed for the 2014 world cup final n defined by f a the jersey number of a is injective. Spure injective modules coincide with cyclically pure exact sequences and cyclically pure injective modules, respectively. The next two chapters contain the main results of the book. Since z is a pid injective z modules are divisible z modules i.
Then there is an embedding m g, where gis an injective rmodule. The following are some facts related to injections. Injective modules and prime ideals of universal enveloping. Then there is f 2ax, monic with all other coe cients in a, such. Projectiveinjective modules, serre functors and symmetric. I have examined the final electronic copy of this thesis for form and content and. For example, a new proof is obtained for the fact that a ring of left. A short exact sequence of amodules is a sequence of the form 0. If m is an injective module over a commutative noetherian ring r, the following conditions are equivalent. Let rbe a commutative noetherian ring with an identity element.
Also, several elementary results will be presented in this section, to ease reading the remainder of the paper. Section 4 is devoted to some applications of our previous results. Yousif dedicated to professor robert wisbauer on his 65th birthday. Then it is reasonable to consider only the epimorphisms, thus we have to take c p2 n2. Modules with cosupport and injective functors 547 above it was tacitly used that if m is a direct summand of hom rc,e,wheree is injective, then m also has this form. Septimiu crivei injective modules relative to torsion. Recall the following characterization of indecomposable injective modules. Projective and injective modules play a crucial role in the study of the cohomology of representations. R module is always an isomorphism if and only if r is odimensional, that. If r is a principal ideal domain, then an rmodule m is said to be. 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. In 4 we proved that injective classes consisting of injective modules are in bijection with certain sets of ideals in r, which we called saturated. Every injective submodule of an injective module is a direct summand, so it is important to understand indecomposable injective modules, lam 1999, 3f. The concept of a module m being almost n injective, where n is some module, was introduced by baba 1989.
An injective module over r r is an injective object in r mod r mod. Mstable under the action of ris caled a submodule of m. Injectives 4 ifmisanr module,theningeneralwemayhaveasemicompletediagramasin lemma2. We now want to introduce a very useful piece of notation that can be used to deal with all these concepts in a uni. For example, all free modules that we know of, are projective modules.
Every indecomposable injective module has a local endomorphism ring. Our main result is a characterization and a description of such classes in terms of ideals. If p is a projective r module over a commutative ring r, then ppiis cpure projective for every ideal i of r. Then you manipulate and simplify, and try to rearrange things to get the right. These are the rings in which every module is injective. Then any injective endomorphism of a finitely generated. Injective hulls of simple modules over some noetherian rings. Stable torsion theories and the injective hulls of simple modules 93 proof.
Oct 28, 2011 examples of projective modules posted. The solution appears to rely on the reasoning of, but it isnt clear why one could reason with dimensions this way. For example, if your ring is not reduced, then the zero module is the only divisible module. Using this class of bck modules, we introduce weakly injective bck modules. It is also proven that a commutative ring r is reduced and. In our approach, projective injective modules, that is modules which are both, projec.
Sam evens these notes are supplementary to class discussion, and much of them is in ash, 10. The study of gorenstein dimensions takes cues from the classical situation. Chinese remainder theorem, endomorphisms of projective modules, projective but not free, tensor product of projective modules. 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. Modules finite generation and ideals m is nitely generated as module,m quotient of ar. R r is one whose graph is never intersected by any horizontal line more than once. If ris a pid, jis an injective rmodule and kis a submodule of jthen jkis injective. Studies of the homology and cohomology modules from the viewpoint of injective objects are often restricted to some subcategories of the category of modules, such as the category of graded or multigraded modules see for example 4, 10, 11 or the category of squarefree modules. 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. A unitary module f over a ring rwith identity, which satis es the equivalent conditions of theorem 4 is called a free rmodule on the set x.
Since every function is surjective when its codomain is restricted to its image, every injection induces a bijection onto its image. This function g is called the inverse of f, and is often denoted by. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. How to understand injective functions, surjective functions. It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is fp injective. We know it is both injective see example 98 and surjective see example 100, therefore it is a bijection. I am submitting herewith a thesis written by ryan neil campbell entitled injective modules and divisible groups.
When r is commutative, the category mr of r modules also has an associative, commutative, and unital smash product, and. We will explore some of these properties in the next. Note that this is equivalent to saying that f is bijective iff its both injective and surjective. Chapter 10 functions nanyang technological university. Projective and injective modules arise quite abundantly in nature. If a function does not map two different elements in the domain to the same element in the range, it is onetoone or injective. Injective modules and prime ideals of universal enveloping algebras. This follows from this existence of injective envelopes. In order to establish this result, we briefly investigate injective modules over a principal ideal domain. Lams textbook contains many examples of injective modules.
Zhou 5, and some wellstudied rings are characterized using simpledirect injective. I suggest you consider only nonzerodivisors, or and probably better you take the definition given in lams book mentioned by yves. It is reasonable to expect that the injective modules over a principal ideal domain also have a simple structure. X y is injective if and only if x is empty or f is leftinvertible. For r r a ring, let r r mod be the category of r r modules. Bijection, injection, and surjection brilliant math. Since pureinjective modules are a generalization of injective modules, it is natural to ask whether the tensor product of pureinjective modules is pureinjective. Then they proved that an nstrongly gorenstein projective module. Y the simple injective module of the nkronecker quiver. The key point of baers argument is to repeat this procedure trans. Here the vertical map is injective, and the diagram commutes. The concept of simpledirect injective modules was introduced by v.
Remember that injective functions dont mind whether some of b gets left out. We know it is both injective see example 98 and surjective see example 100, therefore it is a. Injective modules over such a domain enjoy a divisibility property that is equivalent to injectivity. Module examples i vector spaces are f modules i rings are r modules i abelian groups are z modules i ideals are r modules matrices over a ring 1 i rring, m nr set of n n matrices over r i racts on m nr by scalar multiplication, r7. If it has a twosided inverse, it is both injective since there is a left inverse and surjective since there is a right inverse. Change of basis, transition matrices, and similar matrices 23 6.
Projective and injective modules 29 finally suppose that am. Some generalizations of them were studied by the authors in 11, 14. A wellknown example of a right self injective ring which is not left self injective is the full ring of linear transformations of an infinite dimensional right vector space over a division ring, with linear trans. Exact sequences in the last chapter we have studied many structures related to modules, such as submodules, quotient modules, and module homomorphisms together with their images and kernels. Foundations of module and ring theory a handbook for study and research robert wisbauer. Beachy department of mathematical sciences northern illinois university dekalb, il 60115 for a left rmodule m, we identify certain submodules of mthat play a role analogous to that of prime ideals in the ring r.
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. Module examples i vector spaces are fmodules i rings are rmodules i abelian groups are zmodules i ideals are rmodules matrices over a ring 1 i rring, m nr set of n n matrices over r i racts on m nr by scalar multiplication, r7. A r module eis called injective if for each injective homomorphism f. Every injective module is uniquely a direct sum of indecomposable injective modules, and the indecomposable injective modules are uniquely identified as the injective hulls of the quotients rp where p varies over the prime spectrum of the ring. Avramov abstract examples are given to show that the support of a complex of modules over a commutative noetherian ring may not be read o. Alternatively, f is bijective if it is a onetoone correspondence between those sets, in other words both injective and surjective.
For a given module m, the class of modules n, for which m is almost n injective, is not. R, fx 4x 1, which we have just studied in two examples. Modules and comodules trends in mathematics, 3329 c 2008 birkh. We assume throughout the paper that ris an associative ring with identity, and that rm is a xed left r module. An important example of bijection is the identity function. Functions can be injections onetoone functions, surjections onto functions or bijections both onetoone and onto. Then mhas an essential socle and so its injective hull em is a direct sum of injective hulls of simple left r modules because ris noetherian. Let r r be a commutative ring and c r mod c r mod the category of r r modules. When r is commutative, the category mr of r modules also has an associative, commutative, and unital smash product, and its derived category dr has.
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