A ring r is called a right almost v ring if every simple rmodule is almost injective. Osaka university and osaka city university, departments of mathematics editorial board. Projective and injective modules play a crucial role in the study of the cohomology of representations. On almost projective and almost injective modules let a be an artin algebra, and consider the category moda of. A short exact sequence of amodules is a sequence of the form 0. Almost relative projective modules and almost relative injective modules, monograph. The class of such modules is enveloping and falls properly between injectives and pure injectives. The concept of a module m being almost n injective, where n is some module, was introduced by baba 1989. We give some new properties of almost injective modules and their endomorphism rings, and also provide conditions as to when a direct sum of almost injective or cs modules is again almost. We characterize right noetherian rings over which all simple modules are almost injective. In a nonsemisimple representation theory there are certain spaces associated to homam,ncalled extension groups exti am,n. An exact sequence 0 a f b g c 0ina is an almost split sequence if f is a left almost split morphism and g is a right almost split morphism. The regular and ifinite rings over which every right module is almost projective are described. Almost finjective modules and almost flat modules 117 6 m is an almost pure submodule of its injective envelope em.
It is shown that there exists an rmodule m which is almost injective relative to r but not almost injective. Theorem 3 for any ring r, the category of rmodules has enough injectives. Similarly, the group of all rational numbers and any vector space over any eld are examples of injective modules. Every indecomposable injective module has a local endomorphism ring. We describe rings over which every right module is almost injective. For r r a ring, let r r mod be the category of r rmodules. The structure of rings over which every right module is almost injective is clarified. A module m in moda is called almost projective if ext1 a m,x 6 0 for at most. It is proved that r is a right almost v ring if and only if for every rmodule m, any complement of every. A module m that is almost minjective, is called an almost selfinjective module. Using these concepts, we provide some new characterizations of vrings, semilocal rings and pseudofrobenius rings. 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.
Harada, on almost relative injectives on artinian modules, osaka j. For a module m, the class of those modules n for which m is almost n injective, is not closed under direct sums. We prove that a ring r is semisimple if and only if the rmodule rr is falmost injective. Suppose there is a be an injective rmodule j with the property that h jm 6 0 whenever m6 0. For a given module m, the class of modules n, for which m is almost nm, the class of modules n, for which m is almost n. Almost injective modules lack a baerlike criterion. Noetherian rings with almost injective simple modules.
An almost property of a map gets sent to the actual property by j. A module is called coneat injective if it is injective with respect to all coneat exact sequences. Indecomposable decompositions of finitely presented pureinjective modules. Sam evens these notes are supplementary to class discussion, and much of them is in ash, 10. Projective and injective modules arise quite abundantly in nature. In the language of faithful dimension this means that no injective module m, with m pdf. A rmodule eis called injective if for each injective homomorphism f. Ssinjective modules and rings iraq academic scientific journals. B with b and ba injective, then a is nearly injective itself it has injective dimension 1, and so many arguments for a merely have to combine arguments for b and ba, both of which are supposed to.
Remarks on f p injective and f p flat modules springerlink. An injective module over r r is an injective object in r mod r mod. Uniform almost relative injective modules sciencedirect. We determine the pureinjective hull of every direct sum string module over a string algebra. The concept of a module m being almost ninjective, where n is some module, was introduced by baba 1989. Then there exists an almostinjective function from x to y in particular, the constant function onto y 1 but even though y is t 2, x is not. Pdf a note on almost injective modules adel alahmadi. We introduce the concepts of almostinjective and radinjective modules. A note on almost injective modules 103 proposition 3. Theorem 5 shows that the endomorphism ring of an indecomposable almost selfinjective module is local. Enough injectives university of california, berkeley.
Pdf we give some new properties of almost injective modules and their endomorphism rings, and also provide conditions as to when a direct sum of. M is almost injective if it is almost injective relative to all modules n. We give some new properties of almost injective modules and their endomorphism rings, and also provide conditions as to when a direct sum of almost injective or cs modules is again almost injective or cs in some special cases, n1 is a nonzero. A module m that is almost minjective is called an almost selfinjective module. Pdf almost relative injective modules researchgate. Let r r be a commutative ring and c r mod c r mod the category of r rmodules.
Moreover, the endo morphism ring of a uniserial almost. Relative cotilting theory and almost complete cotilting. M is called almost selfinjective if m is almost minjective. Let rbe a ring with identity and let jbe an rmodule. The author has defined almost projective modules and almost injective modules in 8, and by making use of the concept. Baba gave a necessary and sufficient condition under which a uniform finite length module u is almost vinjective, where v is a finite direct sum of uniform, finite length. It is proved that r is such a ring, if and only if, the complements of. Title almost relative injective modules authors singh. Septimiu crivei injective modules relative to torsion. For a given module m, the class of modules n, for which m is almost ninjective, is not. The concept of a module m being almost ninjective where n is some module, was introduced by baba 1989. For example, all free modules that we know of, are projective modules.
The category a has left almost split sequences if for every indecomposable not injective a. Theorem 5 shows that the endomorphism ring of an indecomposable almost self injective module is local. Baba gave a necessary and sufficient condition under which a uniform. Almost qf rings and almost qr rings manabu harada received june 2, 1992 in this paper we assume that every ring r is an associative ring with identity and r is twosided artinian. Radinjective and almostinjective modules and rings. Injective dimension measures how far from injective a module is. In this note, we study precovers and preenvelopes by f pinjective and f pflat modules, including their properties under almost excellent extensions of rings. If m and n are right rmodules, m is called socleninjective socninjective if every rhomomorphism from the socle of n into m extends to n. Then ris right almost selfinjective if and only if for every c. It is shown that each almost maximal valuation ring r, such that every indecomposable injective rmodule is countably generated, satis.
Almost projective and almost injective modules springerlink. An rmodule jis an injective module if jsatis es one of the equivalent conditions of proposition46. A left rmodule m is called almost finjective, if every rhomomorphism from a finitely presented left ideal to m extends to a homomorphism of. Generalizations of coneat injectivity, like relative coneat injectivity and full invariance of a module in its coneat injective envelope, are studied. Almost finjective modules and almost flat modules 119 4 by proposition 2. In particular, if n is indecomposable, then the diagram 2 gives that for every f. 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. Equivalently, for every semisimple submodule k of n. The articles are devoted to leavitt path algebras, roots of elements in lie groups, hilberts nullstellensatz, mixed multiplicities of ideals, singular matrices, rings of integers, injective hulls of modules, representations of linear, symmetric groups and lie algebras, the algebra of generic matrices and almost injective modules.
Left almost selfinjective rings are defined, similarly. Derived equivalences, restriction to selfinjective. When ndomdima 1, there is a minimal faithful strongly projectiveinjective module ae whose endomorphism ring eae is called the associated selfinjective subalgebra of a. Then the category of rmodules has enough injectives.
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. For a given module m, the class of modules n, for which m is almost ninjective, is not closed under direct sums. A ring r is called right almost selfinjective if it is almost selfinjective as a right module over itself. A module m that is almost m injective, is called an almost self injective module. We provide a negative answer to a question raised by jain and alahmadi asking for a possible baerlike criterion for almost injective modules. Let rbe a commutative noetherian ring with an identity element. A ring r is called right almost selfinjective if it is almost self injective as a right module over itself. We will rst establish the following preliminary result. For a module m, the class of those modules n for which m is almost ninjective, is not closed under direct sums. In addition, we also introduce and investigate f pprojective 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. Pdf a note on almost injective modules researchgate. On almost relative injectives on artinian modules manabu harada received.
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