Every finitelygenerated rmodule is a direct sum of these. In fact, the existence of superdecomposable pure injective modules is a strong re ection of some aspects of the structure of morphisms in. For example, all free modules that we know of, are projective modules. On indecomposable injectives over artinian rings semantic scholar. Let r r be a commutative ring and c r mod c r mod the category of r r 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 sake. Generalov 6, 7 gave a complete description of pureinjective modules over a noetherian serial ring using the structure theory for such rings 11, 17. Journal of pure and applied algebra elsevier journal of pure and applied algebra 119 1997 9153 on indecomposable decompositions of csmodules ii nguyen viet dung institute of mathematics, p. Let r be a noetherian local ring and n be an rmodule, whose elements are annihilated by a power depending on the element of the maximal ideal. In this paper, a sufficient and necessary condition for the indecomposable modules of the self injective algebra of finiterepresentation type to be determined by their composition factors is presented by means of the quiver with relations with the help of covering. Pureinjective and finite length modules over certain rings of differential polynomials gennadi puninski abstract. In these notes i survey the general background to the structure theory of indecomposable. Let r be a noetherian local ring and n be an r module, whose elements are annihilated by a power depending on the element of the maximal ideal.
Every injective module is a direct sum of indecomposable injective modules. In this article we describe the structure of artin algebras a for which any cycle of indecomposable finitely generated right a modules is finite and almost all indecomposable finitely generated a modules do not lie on short paths with injective source and projective target. Mar 27, 20 we classify indecomposable pure injective modules over 1domestic string algebras verifying ringels conjecture on their structure. Finding a precise description of a class of indecomposable injective modules was the main object of 7, 2, 4, 10, 9, and 1, although even over commutative noetherian ring their structure can still be quite complicated. Then the main result of the present paper is the following. Pure injective indecomposable modules over 1domestic string. Finitelygenerated modules over principal ideal domains pids are classified b y the structure theorem for finitely genera ted module s over a principal ideal domain. Loewy series of certain indecomposable modules for. A good example of a structure theorem for a class of indecomposable.
In section 3, we construct injective hulls in terms of generalized fractions. Explicit descriptions of the indecomposable injective modules. A short exact sequence of amodules is a sequence of the form 0. Fisher proved the uniqueness of the representation of a compact module as the pure hull of a direct sum of indecomposable compact short. I show that there is a natural similarity between the structure of the elementary socle series of an indecomposable tt module and the structure of the elementary radical series of its elementary dual. Sam evens these notes are supplementary to class discussion, and much of them is in ash, 10. Furthermore every indecomposable injective module is the injective envelope of a unique simple module. In the category of modules concerning no gradings, a minimal injective resolution of xy consists of in. You have found all indecomposable injective modules. An independence structure on indecomposable modules istwin beck introduction a module a is called an lemodule provided its endomorphism ring is local, and a module m has an ledecomposition if it is isomorphic to a direct sum of le modules. We assume throughout the paper that ris an associative ring with identity, and that rm is a xed left rmodule.
The structure of dualizing complexes is complicated, and is still not well understood, except for some special cases. On injective modules for infinitesimal algebraic groups, ii. The module rx is called minjective if each rhomomorphism f. Pure injective indecomposable modules over 1domestic. Rrp occurring in any direct sum decomposition of a given injective module into indecomposable injectives is independent of the decomposition. The structure of a ring over which every pureinjective resp. An injective module over r r is an injective object in r mod r mod. The indecomposable modules of selfinjective algebra and. It is well known that v is fpinjective, v is the injective hull of v and v. Every finitelygenerated r module is a direct sum of these.
On indecomposable decompositions of csmodules ii sciencedirect. Let r r be a commutative ring and c r mod c r mod the category of r rmodules. Projective and injective modules arise quite abundantly in nature. A rmodule eis called injective if for each injective homomorphism f. We rst show that if mis a nitely generated rmodule, then hom r. For r r a ring, let r r mod be the category of r rmodules. In fact, the existence of superdecomposable pureinjective modules is a strong re ection of some aspects of the structure of morphisms in.
In this paper we study a numerical invariant of dualizing complexes. Rmodule is a direct sum of indecomposable injective r modules. If n0 1 n 2 0 n 3 0 is a chain in f, then in0 i 2fis an upper bound. For example, r is right noetherian if and only if every injective right. E has the following properties, known as matlis duality. Explicit description of a class of indecomposable injective. An independence structure on indecomposable modules. In particular, there are no superdecomposable pureinjective modules over such a ring and every pureinjective indecomposable module is either. A bof rmodules, the associated homomorphism of abelian groups f hom rb,e hom. Selfinjectivity is a moritainvariant property which minimally encompasses frobenius algebras. However we are still short of even formulating a reasonable conjecture on the structure of such modules over nondomestic string algebras. Indecomposable injective modules of finite malcev rank over local commutative rings couchot, francois, journal of commutative algebra, 20.
In most situations, at least, an indecomposable module over an artinian ring has nite length. Explicitly, the modules of the form for prime ideals p including p 0, which yields r are indecomposable. Explicit description of a class of indecomposable injective modules m. Particularly detailed results are obtained when the ring is polynormal.
We study the dimensions of spaces of stable homomorphisms between indecomposable modules which belong to. Some remarks on the structure of indecomposable 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. It follows that every indecomposable right module is pureinjective injective. 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. The structure of the projective modules can be obtained from that of injective modules via the duality functor. Every indecomposable projective module over a semiperfect ring ris isomorphic to refor some primitive idempotent e2r. Due to recent results of harland 8, there is a fair belief that the methods of this paper could be adopted to describe pure injective indecomposable modules over any domestic string algebra. We prove that the indecomposable injective modules have a good. A frobenius algebra is an falgebra atogether with an isomorphism of left modules a. For r r a ring, let r r mod be the category of r r modules. In this paper i show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over.
Similarly, the group of all rational numbers and any vector space over any eld are examples of injective modules. Every indecomposable injective module is the injective hull of the residue field at a prime. Every injective submodule of an injective module is a direct summand, so it is important to understand indecomposable injective modules, lam 1999, 3f. Warfield and fisher defined and developed the structure theory of pure injective modules.
In this note, we describe the explicit structure of the injective envelope of the rmodule rp. In section 2, we study the injective and projective modules in the truncated category. 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 module is called a uniform module if every two nonzero submodules have nonzero intersection.
We obtain existence theorems for extensions, and, for rings satisfying the strong second layer condition, we obtain extensive information about the indecomposable injective modules. Module structure of an injective resolution 3717 x 2. Since a rational injective tg,module is injective on restriction to a sub group scheme h, the indecomposable injective tg,modules all have such filtrations. Explicit descriptions of the indecomposable injective. An independence structure on indecomposable modules istwin beck introduction a module a is called an lemodule provided its endomorphism ring is local, and a module m has an ledecomposition if it is isomorphic to a direct sum of lemodules.
In mathematics, especially in the area of abstract algebra known as module theory, an injective. The elementary socle series has had limited application in describing the structure of certain. In this article we describe the structure of artin algebras a for which any cycle of indecomposable finitely generated right amodules is finite and almost all indecomposable finitely generated amodules do not lie on short paths with injective source and projective target. The kazhdanlusztig combinatorics of translation functors, applied to these projectiveinjective modules, suggests a connection with specht modules for the symmetric group sn. Since the projective indecomposable fgmodules are precisely the direct summands of fg, by. Every indecomposable injective module has a local endomorphism ring. Thomas kucera, department of mathematics, university of. The endomorphism ring of an indecomposable module with an artinian projective cover takeuchi, tsutomu, hokkaido mathematical journal, 1975. Let r be a commutative local noetherian ring with maximal ideal m and. A ring ris noetherian if and only if every direct sum of injective r modules is injective. The number of isomorphic copies of errp occurring in any direct sum decomposition of a given injective module into indecomposable injectives is independent. Projective and injective modules hw pushouts and pullbacks.
We study the dimensions of spaces of stable homomorphisms between indecomposable modules which belong to auslanderreiten components of the form. Injective modules over noetherian rings mathematical sciences. On the number of injective indecomposable modules request pdf. Then there exists a morphism in sji such that 1 is commutative. The kazhdanlusztig combinatorics of translation functors, applied to these projective injective modules, suggests a connection with specht modules for the symmetric group sn. We classify indecomposable pure injective modules over 1domestic string algebras verifying ringels conjecture on their structure. In section 2, we recall the notion of generalized fractions which describe elements in certain top local cohomology modules. On the structure of indecomposable injective modules. Since n is injective, there exists g e, n r such that gif. There is a bijection between the isomorphism classes of simple rmodules and isomorphism classes of projective indecomposable rmodules. Note that this is simple if and only if n 1 or p 0.
Module structure of an injective resolution 3 the paper is organized as follows. For a ring r, the following two conditions are equivalent. Some remarks on the structure of indecomposable injective modules over valuation rings. The structure of indecomposable injective modules 293 inj. In this paper, a sufficient and necessary condition for the indecomposable modules of the selfinjective algebra of finiterepresentation type to be determined by their composition factors is presented by means of the quiver with relations with the help of covering.
Since a rational injective tg,module is injective on restriction to a sub group scheme h, the indecomposable injective tg, modules all have such filtrations. Projectiveinjective modules, serre functors and symmetric. Categories of indecomposable modules i 325 hull of. Projective and injective modules hw pushouts and pull. Matlis 1958 describing the indecomposable injective modules over a commutative noe. If e is an indecomposable injective module over a commutative noetherian ring r,thenassefpgfor some prime ideal p, e is. Some remarks on the structure of indecomposable injective modules over valuation rings article pdf available in communications in algebra 45. Tousi abstract let rbe a commutative noetherian ring and p be a prime ideal of rsuch that the ideal pr p is principal and htp 6 0. The influence of ideal structure on representation theory. Every injective module is uniquely a direct sum of indecomposable injective modules, and the.
Request pdf on the number of injective indecomposable modules for every natural number m, there exists a noncommutative valuation ring r with a completely prime ideal p so that there are. Thomas kucera, department of mathematics, university of manitoba. The kazhdanlusztig combinatorics of translation functors, applied to these projectiveinjective modules, suggests a connection with specht modules for the symmetric group s n. Etrf over such an algebra r is completely determined by the indecomposable projective direct.
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