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