It is pointing out at the outset that pure injective algebraically compact for modules and, indeed, for very general algebraic structures and some. 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. Pdf endomorphism rings of completely pureinjective. Over a nitedimensional algebra a module is pure injective if it is a di. Superdecomposable pureinjective modules over tubular algebras. Generalizations of coneat injectivity, like relative coneat injectivity and full invariance of a module in its coneat injective envelope, are studied. Pdf duality of preenvelopes and pure injective modules. It is well known that over a left notherian ring any direct sum of injecive modules is again injective, and every injective module is a direct sum of inde. 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. Hidetoshimarubayashi received novemver 1, 1979 revised may 15, 1980 the notion of purity in abelian groups is one of the most important types in abelian groups. This property follows directly from the fact that zorns lemma can be used to extend a basis of u to a basis of v. It indicates that the intersection of the category of pure injective modules and that. The class of all pure injective modules is denoted by pi. Pure injective indecomposable modules over 1domestic string.
Pdf endomorphism rings of completely pureinjective modules. Pure injective modules relative to torsion theories 189 quences 0. Rings of pure global dimension less than or equal to onerecall that the pure injective dimension of a right r module m is defined as the smallest nonnegative integer or. Moreover, every pure injective right module over ris the pure injective envelope of a direct sum of indecomposable modules. In addition, we get that the class of rd injective left r modules is closed under extensions if and only if every war. A ring r is noetherian if and only if every absolutely pure r module is injective. We first characterize submodules that have the extension property with respect to copure injective modules. The existence of relative pure injective envelopes s pure exact sequences the notion ofpuritywas introduced bycohn1959forleft rmodulesand bylos. We show that every module can be embedded as qcopure submodule of a qcopure baer injective module. Superdecomposable pureinjective modules over tubular. Modr is pure projective if it is projective with respect to pure exact sequences, that is, any pure exact sequence 0 a b m 0 in modr ending at m splits. R is the pure injective envelope as a right r module of a direct sum of pure injective indecomposable right r modules b i.
Using similar techniques, it is also shown that an r module has its underlying group pure injective precisely when it is a submodule of a product of duals of cyclic modules and also a summand as abelian group of the same product. All known examples of cotilting modules are pure injective. Let mbe an r module and sa multiplicatively closed set of r. Let r be a commutative ring with identity and d an rmodule. If n is a pure submodule of m, then m is a pure essential extension of n if there are no nonzero submodules xm with xfn0 and n\xx pure in mjx.
The class of such modules is enveloping and falls properly between injectives and pure injectives. Finally, we prove that the following are equivalent for a ringr and an integer n. Jan 24, 2002 a pure projective module is l pureinjective if and only if it is a direct sum of countably generated pure projective, pureinjective submodules. Let r be a commutative ring with identity and d an r module. Superdecomposable pure injective modules over commutative. In all of the examples of rings cited above, the pure injective envelope of r r is a. Pure injective modules relative to torsion theories.
The solutions to these systems allow the extension of certain kinds of module homomorphisms. We show that some characterizations similar to those for injective complexes exist for fp injective complexes. In mathematics, 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. Sep 17, 2010 a module is said to be copure injective if it is injective with respect to all modules a. Pdf pureinjective modules over tubular algebras and string. A module m is called pure injective if for any module a and any pure submodule b of a y any homomorphism. So we would like to learn whether there exist such projective and injective modules that are in some sense minimal and universal for m, sort of closures for mamong the projective and injective modules. It is proved that a pure injective module over a commutative ring with unity is a summand of a product of duals of finitely presented modules, where duals are to be understood with reference to the circle group t, with induced module structures. An r module a is absolutely pure if and only if extgaf, a 0for all finitely presented r modules m. It turns out that a module is algebraically compact if and only if it is pure injective. Modr is absolutely pure if any exact sequence 0 m b c 0 in modr starting from m is pure exact. A submodule aof a module bis pure a b, for short if for each. Modr is called pure injectiveif it is an injective object with respect to pureexact sequences.
A module m is called subinjective relative to a module n if for every extension k of. Pureinjectivity in the category of gorenstein projective. If u is a subspace of a vector space v over a division ring, then u is a direct summand of v. Over a left noetherian ring each absolutely pure left a module is injective. Toinvestigate such algebraic structures wehad to define what wecall pessential extension modules, pure relative complement submodules, left pure hereditary tings and someother related notions. The subject of this article are the modules m over a ring r such that every element of m is contained in a pure injective direct summand of m. In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module q that shares certain desirable properties with the z module q of all rational numbers. Then by assuming that the ring is noetherian or pure semisimple we show that all limit models are injective see theorem. Many authors defined the concept of purity in modules over noncommutative rings and generalized some of well known results in pure injective abelian groups to the case of modules. The lattice of pp formulas is naturally isomorphic to that of pointed nitely presented modules, that is pairs m. We also introduce and study the notion of an fp injective dimension associated to every complex of left r modules over an arbitrary ring. Pdf pureinjective modules over tubular algebras and.
A short exact sequence of a modules is a sequence of the form 0. Every module can be imbedded as a qcopure submodule in some qcopure baer injective module. Specifically, if q is a submodule of some other module, then it is already a direct summand of that module. First, we show that long limit models in the class of modules with embeddings are injective modules see lemma 3. This observation and corollary 1 together yield theorem 3. Since a is torsion free, a is also imbedded in ete, which is torsion free and divisible. Some properties and some characterization of such modules are established. Pure projective modules and fpinjective modules over morita. Let p be a projective r module and let i be an ideal of r and. All modules with finitely many elements are algebraically compact. Pureinjective modules prest, mike 2008 mims eprint mims eprints. The frobenius functor and injective modules 3 proposition 2.
As an alternative perspective on the injectivity of a pure injective module, a pure injective module m is said to be piindigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. 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. In this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure baer injective module. Pure injective modules over hereditary noetherian prime rings with enough invertible ideals. The module m is pure injective if any pure injective homomorphism j. Depending on the class q, this concept is both a dualization and a generalization of pure baer injectivity. A classical question of yoneda asks when the tensor product of two injective modules is injective. Works of angelerihugel, enochs, krause, rada, and saorin make it easy to discuss covering and enveloping properties of modrb, and furthermore we compare the naturally associated notions of bcoherence and bnoetherianness. Then we characterize commutative rings with finite self injective dimension in terms of copure injective modules. Pureinjective modules prest, mike 2008 mims eprint. Pdf some decomposition properties of injective and pure. We also introduce two notions closely related to pure baer injectivity. These algebraically compact modules are analogous to injective modules, where. 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 pure injective modules of slope rhas mdimension 2 if ris rational, and unde ned breadth if ris irrational and hence that there are no.
There are conditions on a ring which guarantee that the class of injective modules will be covering. If one chooses a basis of ete, then each of the generators of a is a linear combination of only finitely many basis vectors. In this paper the analogue question for pure injective modules is studied. Pdf in this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure baer injective module. If n is a pure submodule of m, then m is a pure essential extension of n if there are no nonzero submodules xm with xfn0 and n\xx pure.
Aring ris an ssbiring if and only ifevery smisimple rmoduleis pure baerinjective. We will prove that r is left noetherian and right perfect cf. Modules that are injective with respect to pure embeddings are called pure injective. Enochs proved in 4 that if r is a commutative noetherian ring, then homrm, e is a pureinjective rmodule for any rmodules m and e with e. Jun 18, 20 absolutely pure modules act in ways similar to injective modules. Pure projective modules and fpinjective modules over. Aug 02, 2018 for a given class of r modules q, a module m is called qcopure baer injective if any map from a qcopure left ideal of r into m can be extended to a map from r into m. Pureinjectivity of tensor products of modules algebra. A submodule a of a left module b over an associative ring with 1 is pure if for any right module f, the natural homomorphism f. Applications of duality to the pureinjective envelope. If the class cogenw is closed under direct limits, then w is pure injective. T 1 modules are the indecomposable nitedimensional modules, the prufer modules, which are the injective envelopes of the simple modules, and the function eld kt. Imbedding in copure baer injective modules the main result of this section is the following. Pureinjective modules glasgow mathematical journal.
A module is called coneat injective if it is injective with respect to all coneat exact sequences. If one chooses a basis of ete, then each of the generators of a is a linear combination of only finitely many. We determine the pure injective hull of every direct sum string module over a string algebra. We show that there is a close connection between the fp injective dimension of complexes and flat. Baur proved that for any fixed module a4 every formula is equivalent to a boolean combina.
Bull acad polon sci ser sci math astronom phys, 1967, 15. It is shown that if d is pure injective, then d is isomorphic to a direct. Pdf 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. All sums and products will be taken over the index set 7. A complete answer to this question was given by enochs and jenda in 1991. Pdf on relative pure injective modules researchgate. It is shown that if d is pure injective, then d is isomorphic to a direct summand. Then we introduce the rdprojective and rd injective dimensions for modules and. Pure injective indecomposable modules over 1domestic. It is pointing out at the outset that pureinjective algebraically compact for modules and, indeed, for very general algebraic structures and some. Many authors defined the concept of purity in modules over noncommutative rings and generalized some of well known results in pure injective abelian groups to the case of modules cf.
Thus if r is not noetherian, we can form a direct sum of injective. 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. An r module d is said to be s pure injective if for any s pure exact sequence 0 ab c 0, the induced homomorphism homrb,dhomra,d is surjective. Mar 27, 20 we classify indecomposable pure injective modules over 1domestic string algebras verifying ringels conjecture on their structure. Properties of rdprojective and rd injective modules lixin mao abstract in this paper, we.
Mao left r module if and only if m is injective relative to every rdexact sequence 0 k p l 0of left r modules with prdprojective. Pureinjectivity in the category of gorenstein projective modules. Pure injective modules over hereditary noetherian prime rings. A module rw is pure injective if it is injective with respect to any pure exact sequence. It naturally arises the question how the two notions are related. Introduction the purpose of this paper is to give a complete description of pure injective modules over a serial right noetherian ring. If p is a projective r module over a commutative ring r, then ppiis c pure projective for every ideal i of r. In fact a powerful criterion of ziegler 15, theorem 7.
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