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. 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. 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. 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. The solutions to these systems allow the extension of certain kinds of module homomorphisms. Applications of duality to the pureinjective envelope. Toinvestigate such algebraic structures wehad to define what wecall pessential extension modules, pure relative complement submodules, left pure hereditary tings and someother related notions. Pdf pureinjective modules over tubular algebras and. Thus if r is not noetherian, we can form a direct sum of injective. Superdecomposable pureinjective modules over tubular. If one chooses a basis of ete, then each of the generators of a is a linear combination of only finitely many. An r module a is absolutely pure if and only if extgaf, a 0for all finitely presented r modules m.
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. Pure projective modules and fpinjective modules over morita. The module m is pure injective if any pure injective homomorphism j. Specifically, if q is a submodule of some other module, then it is already a direct summand of that module. 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. 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. Pdf pureinjective modules over tubular algebras and string.
Pureinjective modules glasgow mathematical journal. R is the pure injective envelope as a right r module of a direct sum of pure injective indecomposable right r modules b i. Properties of rdprojective and rd injective modules lixin mao abstract in this paper, we. In this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure baer injective module. 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. It is pointing out at the outset that pureinjective algebraically compact for modules and, indeed, for very general algebraic structures and some. Jun 18, 20 absolutely pure modules act in ways similar to injective modules. A submodule aof a module bis pure a b, for short if for each. 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. 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. It is shown that if d is pure injective, then d is isomorphic to a direct.
In fact a powerful criterion of ziegler 15, theorem 7. It indicates that the intersection of the category of pure injective modules and that. This observation and corollary 1 together yield theorem 3. Modules that are injective with respect to pure embeddings are called pure injective. It is pointing out at the outset that pure injective algebraically compact for modules and, indeed, for very general algebraic structures and some. Pdf endomorphism rings of completely pureinjective. A complete answer to this question was given by enochs and jenda in 1991.
Pure injective indecomposable modules over 1domestic. A short exact sequence of a modules is a sequence of the form 0. If p is a projective r module over a commutative ring r, then ppiis c pure projective for every ideal i of r. Pure injective modules over hereditary noetherian prime rings. 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. 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. 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. Aring ris an ssbiring if and only ifevery smisimple rmoduleis pure baerinjective. Pure injective indecomposable modules over 1domestic string. It turns out that a module is algebraically compact if and only if it is pure injective. Pureinjectivity in the category of gorenstein projective modules. The frobenius functor and injective modules 3 proposition 2.
Depending on the class q, this concept is both a dualization and a generalization of pure baer injectivity. The class of such modules is enveloping and falls properly between injectives and pure injectives. 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. 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. 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. Let mbe an r module and sa multiplicatively closed set of r. This property follows directly from the fact that zorns lemma can be used to extend a basis of u to a basis of v. We show that every module can be embedded as qcopure submodule of a qcopure baer injective module. 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. All modules with finitely many elements are algebraically compact. Pdf duality of preenvelopes and pure injective modules.
Then we introduce the rdprojective and rd injective dimensions for modules and. Pdf endomorphism rings of completely pureinjective modules. The class of all pure injective modules is denoted by pi. 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. A module rw is pure injective if it is injective with respect to any pure exact sequence. If u is a subspace of a vector space v over a division ring, then u is a direct summand of v. Pureinjective modules prest, mike 2008 mims eprint mims eprints. There are conditions on a ring which guarantee that the class of injective modules will be covering. A module m is called pure injective if for any module a and any pure submodule b of a y any homomorphism. Pureinjective modules prest, mike 2008 mims eprint.
Generalizations of coneat injectivity, like relative coneat injectivity and full invariance of a module in its coneat injective envelope, are studied. The existence of relative pure injective envelopes s pure exact sequences the notion ofpuritywas introduced bycohn1959forleft rmodulesand bylos. Modr is called pure injectiveif it is an injective object with respect to pureexact sequences. All known examples of cotilting modules are 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. We determine the pure injective hull of every direct sum string module over a string algebra. Modr is absolutely pure if any exact sequence 0 m b c 0 in modr starting from m is pure exact.
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. Then by assuming that the ring is noetherian or pure semisimple we show that all limit models are injective see theorem. In addition, we get that the class of rd injective left r modules is closed under extensions if and only if every war. We show that some characterizations similar to those for injective complexes exist for fp injective complexes. First, we show that long limit models in the class of modules with embeddings are injective modules see lemma 3. In this paper the analogue question for pure injective modules is studied.
A ring r is noetherian if and only if every absolutely pure r module is injective. A module m is called subinjective relative to a module n if for every extension k of. These algebraically compact modules are analogous to injective modules, where. Every module can be imbedded as a qcopure submodule in some qcopure baer injective module. Let r be a commutative ring with identity and d an r module. Pureinjectivity in the category of gorenstein projective. We also introduce and study the notion of an fp injective dimension associated to every complex of left r modules over an arbitrary ring. 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. In all of the examples of rings cited above, the pure injective envelope of r r is a. The lattice of pp formulas is naturally isomorphic to that of pointed nitely presented modules, that is pairs m. 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. Pureinjectivity of tensor products of modules algebra. We first characterize submodules that have the extension property with respect to copure injective modules. Then we characterize commutative rings with finite self injective dimension in terms of copure injective modules.
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. All sums and products will be taken over the index set 7. Superdecomposable pureinjective modules over tubular algebras. 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. Over a nitedimensional algebra a module is pure injective if it is a di. Pure injective modules relative to torsion theories 189 quences 0. Pdf some decomposition properties of injective and pure. A module is called coneat injective if it is injective with respect to all coneat exact sequences. Since a is torsion free, a is also imbedded in ete, which is torsion free and divisible. 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. A classical question of yoneda asks when the tensor product of two injective modules is injective. 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. Some properties and some characterization of such modules are established. Bull acad polon sci ser sci math astronom phys, 1967, 15.
Pure projective modules and fpinjective modules over. Moreover, every pure injective right module over ris the pure injective envelope of a direct sum of indecomposable modules. Superdecomposable pure injective modules over commutative. We will prove that r is left noetherian and right perfect cf. Over a left noetherian ring each absolutely pure left a module is injective. If one chooses a basis of ete, then each of the generators of a is a linear combination of only finitely many basis vectors. If the class cogenw is closed under direct limits, then w is pure injective. Let r be a commutative ring with identity and d an rmodule. Pdf on relative pure injective modules researchgate. Finally, we prove that the following are equivalent for a ringr and an integer n.
Let p be a projective r module and let i be an ideal of r and. Pure injective modules over hereditary noetherian prime rings with enough invertible ideals. Imbedding in copure baer injective modules the main result of this section is the following. Pure injective modules relative to torsion theories. Pdf in this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure baer injective module. 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. 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.
Introduction the purpose of this paper is to give a complete description of pure injective modules over a serial right noetherian ring. Sep 17, 2010 a module is said to be copure injective if it is injective with respect to all modules a. An r module j is injective if for any exact sequence 0. 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.
We also introduce two notions closely related to pure baer injectivity. We show that there is a close connection between the fp injective dimension of complexes and flat. It is shown that if d is pure injective, then d is isomorphic to a direct summand. 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. Mar 27, 20 we classify indecomposable pure injective modules over 1domestic string algebras verifying ringels conjecture on their structure. Baur proved that for any fixed module a4 every formula is equivalent to a boolean combina. It naturally arises the question how the two notions are related.
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