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. Infinitary axiomatizations lead to coherence results previously known for the special case of. Also, several elementary results will be presented in this section, to ease reading the remainder of the paper. We recall that an rmodule m is called pureinjective if all direct sums of copies of m are pureinjective. We prove that over every nondomestic string algebra over a countable eld there exists a superdecomposable pureinjective module. As preparation, descriptions of pure projective and mittagleffler preenvelopes with respect to socalled definable subcategories and of pure generation for such are derived, which may be of interest on their own. Nowri, being the direct sum oftwo pure baer injective r modules, should be pure baer injective. S pure injective modules coincide with cyclically pure exact sequences and cyclically pure injective modules, respectively. F 0g, where f is a nite subset of m, is a base for the lter of neighbourhoods of zero. Firstly any jfpure injective module is a direct sum of an injective module and an.
It is proved in this paper that an r module m is quasipureinjective iff every pure exact sequence 0 m b splits whenever there is a homomorphism. In fact, as proved in 1 and 9, anrmodule uis cotilting if and only if the following. Then we get an equivalent characterization of virtually gorenstein algebras being cm nite. Pureinjective envelopes journal of algebra and its. Pdf relative injectivity and pureinjective modules over prufer. Superdecomposable pureinjective modules over tubular.
It is proved that if c is a locally finitely presented additive category, then every object x of c admits a pure injective envelope. We show that, for any tubular algebra, the lattice of ppdefinable subgroups of the direct sum of all indecomposable pure injective modules of slope r has mdimension 2 if r is rational, and undefined breadth if r is irrational and hence that there are no superdecomposable pure injectives of rational slope, but there are superdecomposable pure injectives of irrational slope, if the underlying. We prove diverse characterizations, some structural results and give conditions under which locally pureinjectives are pureinjective. As ris left noetherian, fpinjective rmodules are injective. Now, we are in a position to define a new class of modules which contains the class of pureinjective modules but is larger in general. It is proved in this paper that an r module m is quasi pure injective iff every pure exact sequence 0 m b splits whenever there is a homomorphism. Pure injective modules relative to torsion theories. A module is called coneat injective if it is injective with respect to all coneat exact sequences. It is shown that if d is pure injective, then d is isomorphic to a direct summand. It is also interesting to study in nitedimensional modules, especially pure injective modules, see 12, 9, 10.
We describe all those indecomposable pureinjective rmodules m with finitedimensional top, that is, such that mradrm is finite dimensional over rradr. Proceedings conference on abelian groups and modules, udine 1984, 383393 wiennew york 1984. On locally pureinjective modules, journal of pure and. In this paper we classify indecomposable pure injective modules for string algebras. As application of these results we obtain again the structure of indecomposable pureinjective modules over a valuation domain, showing that their elements have constant indicator. 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. The question is addressed of when all pureprojective modules are direct sums of finitely presented modules. Pdf in this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure baer injective module. Recall that a module is said to be pure injective or algebraically compact if it is injective with respect to pure. Certain properties of quasi injective and absolutely pure modules are extended to absolute self purity. If p is a projective rmodule over a commutative ring r, then ppiis c pure projective for every ideal i of r. 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. Maybe the most important injective module is the abelian group qz.
N is said to be pure if for any rmodule l, the map f. It is proved that this is the case over hereditary noetherian rings. Prest proved that in this case the width of the lattice of all pointed amodules is undefined and hence there exists a superdecomposable pureinjective amodule, if the base field k is countable. 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. One of them is the notion of pure injective modules. Pure injective modules over tubular algebras and string algebras january 30, 2011 we show that, for any tubular algebra, the lattice of ppde nable subgroups of the direct sum of all indecomposable pure injective modules of slope rhas mdimension 2 if ris rational, and unde ned breadth if ris irrational and hence that there are no. Considered as a morphism, the pureinjective envelope is a pure. Aleft rmodulemis called puresplit ifeverypure submodule ofmis a direct summand.
For obvious reasons we call these modules locally pureinjective. Cotilting versus pureinjective modules 5 let us endow 2m with the nite topology. For instance, the cantor bendixson rank of the ziegler spectrum over d1 is equal to 2 and there is no superdecomposable pureinjective module. Jun 18, 20 absolutely pure modules act in ways similar to injective modules.
Pure injective modules over hereditary noetherian prime. Some of the methods are modeltheoretic, and the techniques developed using these may be of interest in their own right. Introduction there is a wellknown dichotomy for the behavior of a. We prove diverse characterizations, some structural results and give. All the known results support the conjecture of prest concerning superdecomposable pureinjective modules. Since pure injective modules are a generalization of injective modules, it is natural to ask whether the tensor product of pure injective modules is pure injective. In most of these papers it is assumed that the base. It is pointing out at the outset that pureinjective algebraically compact for modules and, indeed, for very general algebraic structures and some. Nowri, being the direct sum oftwo pure baer injective rmodules, should be pure baer injective. Absolutely pure modules act in ways similar to injective modules.
Chapter v pureinjective modules a module is called pureinjective if it is a direct summand of every module in which it is a pure submodule. Furthermore, we show that the sets of matrix subgroups of the modules in question satisfy the ab5. Pureinjective modules prest, mike 2008 mims eprint. The purpose of this survey is to present and illustrate the definitions and a number of the results around pureinjective modules. The purpose of this survey is to present and illustrate the definitions and a number of the results around pure injective modules. Pure injective modules over hereditary noetherian prime rings. We show that, for any tubular algebra, the lattice of ppdefinable subgroups of the direct sum of all indecomposable pureinjective modules of slope r has mdimension 2 if r is rational, and undefined breadth if r is irrational and hence that there are no superdecomposable pureinjectives of rational slope, but there are superdecomposable pureinjectives of irrational slope, if the underlying. Some results on pure injective modules over a commutative ring with 1, proved by ziegler using model theory, are proved here through algebraic methods. This yields r is countable and hence one can apply the criterion of ziegler. Pdf cotilting versus pureinjective modules francesca. Pointed modules and morphisms between them, as well as the essentially equivalent pp formulas and associated methods from the model theory of.
Matlis 1958 describing the indecomposable injective modules over a commutative noetherian ring. Thus, is degreewise an exact sequence of injective modules, in particular it is degreewise split exact. Superdecomposable pureinjective modules over tubular algebras mike prest, university of manchester, manchester, uk september 11, 2012 i report some results from the doctoral thesis 2 of my student richard harland. The class of such modules is enveloping and falls properly between injectives and pure injectives.
In fact, as proved in 1 and 9, anrmodule uis cotilting if and only if the following three conditions hold. Assume that k is an algebraically closed field and a is a finitedimensional wild kalgebra. The structure of a ring over which every pureinjective resp. In this direction, the following two theorems give us conditions which guarantee that the tensor product of two pureinjective modules is pureinjective. When cotorsion modules are pure injective journal of.
Pureinjective modules prest, mike 2008 mims eprint mims eprints. Pureinjective modules over tubular algebras and string. Preprint versions are available above in pdf format. Considered as a morphism, the pure injective envelope is a pure. Aleft rmodulemis called pure split ifeverypure submodule ofmis a direct summand. Let p be a projective rmodule and let i be an ideal of r and. Generalizations of coneat injectivity, like relative coneat injectivity and full invariance of a module in its coneat injective envelope, are studied. In theorem 16, we prove that ccan be any class of cotorsion modules when r is a dedekind domain. There are conditions on a ring which guarantee that the class of injective modules will be covering. Superdecomposable pureinjective modules exist over some string algebras gena puninski abstract. It obviouslyfollowsthat everyleft puresplitringis left purehereditary.
Pure injective modules relative to torsion theories 189 quences 0. Now, we are in a position to define a new class of modules which contains the class of pure injective modules but is larger in general. Spure injective modules coincide with cyclically pure exact sequences and cyclically pure injective modules, respectively. Infinitary axiomatizations lead to coherence results previously known for the special case of flat modules. In this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure baer injective module. One might ask how far such a structure theory might be extended to arbitrary indecomposable. Since pureinjective modules are a generalization of injective modules, it is natural to ask whether the tensor product of pureinjective modules is pureinjective.
Pureinjectivity in the category of gorenstein projective. Furthermore, we prove that the category of weak pureinjective gorenstein projective left rmodules is enveloping in the category of left rmodules. Pdf duality of preenvelopes and pure injective modules. Furthermore, we show that the sets of matrix subgroups of the modules in. Finite length and pureinjective modules over a ring of. We describe all those indecomposable pure injective r modules m with finitedimensional top, that is, such that mradrm is finite dimensional over rradr. All modules with finitely many elements are algebraically compact.
Now the assertion follows from the fact that the class of pureinjective modules is closed under direct summands. On locally pureinjective modules on locally pureinjective modules zimmermann, wolfgang 20020124 00. Some properties and some characterization of such modules are established. Let r be the pullback, in the sense of 9, of two dedekind domains. It obviouslyfollowsthat everyleft pure splitringis left purehereditary. We observe that if rw s is a pure injective cotilting bimodule, both r and s are semiregular rings and we give a characterization of the re exive modules in terms of a suitable. Pdf on relative pure injective modules researchgate. The extension of this theory to modules over arbitrary rings became possible after the work of baur l, monk 14, fisher 6 and warfield 30. If p is a projective rmodule over a commutative ring r, then ppiis cpure projective for every ideal i of r.
Marandas theorem for pureinjective modules and duality. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. The module m is pureinjective if any pure injective homomorphism j. Projective and injective modules play a crucial role in the study of the cohomology of representations. Then we have 1 there exists an isomorphism of le smodules. Model theory of modules, ziegler spectra, injective modules, representations of finite dimensional algebras, ordered fields. Pdf it is proved in this paper that an r module m is quasipureinjective iff every pure exact sequence 0 m b splits whenever there is a.
As application of these results we obtain again the structure of indecomposable pure injective modules over a valuation domain, showing that their elements have constant indicator. Chapter v pure injective modules a module is called pure injective if it is a direct summand of every module in which it is a pure submodule. It is proved that if c is a locally finitely presented additive category, then every object x of c admits a pureinjective envelope. Regular and left noetherian rings are characterized using this new concept. On wild algebras and superdecomposable pureinjective modules. It is defined via the s pure monos in a similar way as the absolutely pure modules are defined via the pure monos. In section 3, we present two characterizations of cyclically pure injective modules. It turns out that a module is algebraically compact if and only if it is pureinjective. Some results on pureinjective modules over a commutative ring with 1, proved by ziegler using model theory, are proved here through algebraic methods. It is defined via the spure monos in a similar way as the absolutely pure modules are defined via the pure monos. Let r be a commutative ring with identity and d an rmodule. Pureinjective modules over tubular algebras and string algebras january 30, 2011 we show that, for any tubular algebra, the lattice of ppde nable subgroups of the direct sum of all indecomposable pureinjective modules of slope rhas mdimension 2 if ris rational, and unde ned breadth if ris irrational and hence that there are no. The subject of this article are the modules m over a ring r such that every element of m is contained in a pureinjective direct summand of m.
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