Example if ris a division ring, then ris morita equivalent to all of its matrix rings m nr. Relative singularity categories and gorensteinprojective modules. A note on endomorphism rings of semiprojective modules article pdf available in mathematical proceedings of the royal irish academy 1122. On the nonrigidity of the frobenius endomorphism over. Endomorphism rings of torsionless modules by arun vinayak jategaonkar abstract. Torsionfree abelian groups of finite rank projective as.
Find all the books, read about the author, and more. These keywords were added by machine and not by the authors. Computing global dimensions of endomorphism rings of. This book presents topics in module theory and ring theory. From now on, all rings will be associative rings with 1 6 0 which may or may not be commutative and rmodule will be understood to be left rmodules. Regular endomorphism rings and principally injective modules. For a commutative local ring r, consider noncommutative ralgebras. To generalize tilting modules over an arbitrary ring and support. The algebra a is the image of the group algebra of g in the endomorphism ring of m. Pdf projective modules with semilocal endomorphism ring. Endomorphism rings of projective modules ams transactions of. An introduction to morita theory school of mathematics. China dedicated to professor zhou boxun for his 80th birthday. For the special case of rank one projective modules, this description presents the di erential picard group in terms of picr and a quotient of the additive group of r.
The collection of all left representations of a ring r, that is, the collection of all left rmodules, forms a very rich and interesting category. Modular representation theory of endomorphism rings. On the other hand, we prove that, a ring r is a right i 1. Note that the regular modules rrand r r are both faithful. Then we give an affirmative answer to a problem of dugas a construction of derived equivalent pairs of symmetric algebras, proc.
This process is experimental and the keywords may be updated as the learning algorithm improves. A non zero module x is called a proper subfactor of a module m if x is a proper. Ellenberg department of mathematics, princeton university fine hall, washington road princeton, nj 08544 email. Projective modules over the endomorphism ring of a.
Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Apart from this, we have introduced the idea of mhollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi projective modules is a local ring. Thus one also has the quotient triangulated category d t a. An endomorphism that is also an isomorphism is an automorphism. Isomorphisms between endomorphism rings of projective modules. Endomorphism rings of permutation modules natalie naehrig lehrstuhl d f ur mathematik rwth aachen university. The second chapter focuses on direct sum decompositions for modules. Over a ring a let pa be a finitely generated projective right a module with aendomorphism ring b. Let r p be a projective r module which is not finitely generated. We give also an explicit formula for the socle of a projective object with a short proof using soergels functor vand nish with a generalization of this functor to harishchandra bimodules and parabolic versions of.
We have discussed the summand intersection property, summand sum property for semi projective modules. Endomorphism rings of finitely generated projective modules. It is well known that, over a serial artinian ring, every indecomposable left and every indecomposable right module is finitely generated serial, quasi projective, and quasiinjective 3,5. It is wellknown that for a large class of local rings of positive characteristic, including complete intersection rings, the frobenius endomorphism can be used as a test for finite projective dimension. Section 2 deals with the interconnection between finite ness condition in module m and its endomorphism ring endmr.
For a ring a, we will let a denote the ring obtained from a by adjoining an identity element in the customary manner, and a will denote the jacobson radical of a. Recall that for a a ring with 1, a left amodule p is projective ii given any homomorphism f. A natural source of local rings is the class of endomorphism rings of indecomposable modules. Over a ring a let pa be a finitely generated projective right a module with a endomorphism ring b. Endomorphism rings and direct sum decompositions in some classes of modules, birkh auser verlag, basel 1998, i stated 21 \open problems. Rings of endomorphisms of semigroupgraded modules abrams, gene and menini, claudia, rocky mountain journal of mathematics, 1996. On the endomorphism ring of the canonical module aoyama, yoichi and goto, shiro, journal of mathematics of kyoto university, 1985. Endomorphisms of finetely generated projective modules. In general, we can talk about endomorphisms in any category. Thus, kaplanskys result cannot be generalized even for rings with two maximal ideals. On the stable range of endomorphism rings of quasi projective modules tongsuo wu department of applied mathematics shanghai jiaotong university shanghai 200030, p. Endomorphism rings and direct sum decompositions in some classes of modules alberto facchini in chapter 11 of my book \module theory.
It is shown that a projective module has a local endomorphism ring if and only if it is a cyclic module with a unique maximal ideal. Introduction all rings in this paper are associative with identity and all modules right unital. However quasibaer modules need not be principally projective. In the second part, we study exchange rings with the right powersubstitution property. Pdf on semi projective modules and their endomorphism rings. Namely, matrix rings over a commutative domain rare quasibaer rings. Endomorphism rings of quasiprojective modules tongsuo wu department of applied mathematics, shanghai jiaotong university, shanghai 200030, p. In this case, we say that ris the ring of formal power series associated to h.
In mathematics, an endomorphism is a morphism from a mathematical object to itself. We generalize a theorem of stock, which shows that projective modules with. Throughout this paper, unless otherwise indicated, all modules over a ring r will be understood to be left amodules. Well see a proof of a special case of this later on, using quivers. It is shown that the endomorphism ring of a finitely generated regular module is a regular ring. In a similar fashion, if r is any commutative ring, the endomorphism monoids of its modules form algebras over r by the same axioms and derivation. Hence it is isomorphic to the quotient of the group algebra kg by the annihilator in kg of the module m. Let r and s be rings and let rgji and sgjl be the categories of all left r. However, as we will show in this paper, the theory of in. Pdf semipotency and the total of rings and modules researchgate.
Rt 25 aug 2015 computing global dimension of endomorphism rings via ladders brandon doherty, eleonore. This paper examines the structure of p as an emodule and relations between right ideals of r and esubmodules of p as an emodule and relations between right ideals of r and esubmodules of. If a ring is embeddable in the endomorphism ring of a. The classi cation of di erential projective modules, like the case of ordinary projective modules, is a question of ktheory, which we describe formally. Pdf this paper provides the several homological characterization of perfect rings and semisimple rings in terms of semiprojective modules. Projective modules with prime endomorphism rings springerlink. In this paper we characterize the torsionfree abelian groups, a, of finite rank which are projective as modules over their endomorphism rings, e. In this paper, we exploit this property to study the structure of such rings. Jondrup 11 has studied when r is projective as an rgmodule. Curves arising from endomorphism rings of kronecker modules. Endomorphism rings of small pseudo projective modules 607 proposition. Recently, some authors studied some properties of rings and modules via regularity and injectivity of modules, homomorphisms. Endomorphism rings of quasi projective modules tongsuo wu department of applied mathematics, shanghai jiaotong university, shanghai 200030, p.
Let m be any pseudo projective module and endm denotes the endomorphism ring of m. Fleury, patrick, pacific journal of mathematics, 1974. Di erential projective modules over di erential rings. Lectures on noncommutative rings university of oregon. Relative singularity categories and gorensteinprojective modules a and a selforthogonal amodule t, we have that kbadd t can be viewed as a triangulated subcategory of dbamod where add t is the full subcategory of amod consisting of direct summands of. Q,s is a weight if and only if sis a simple and projective kngqqmodule.
This paper examines the structure of p as an e module and relations between right ideals of r and esubmodules of. We then write kg p for the permutation kg module on the cosets of p in g, and put e. Directsum decompositions of modules with semilocal endomorphism rings. More generally, a projective module has a semiperfect endomorphism ring if and only if it is a finite direct sum of modules each of which has a local endomorphism ring. Pdf a note on endomorphism rings of semiprojective modules. Natalie naehrig modular representation theory of endomorphism rings. Also torsion modules over a topological ring and graded modules over a graded ring form categories of the type. Mares 9 has called a projective module semiperfect if every homomorphic image has a projective cover and has shown that many of the properties of semiperfect rings can be extended to these modules. Projective modules over the endomorphism ring of a biuniform. Thus kaplanskys result can not be generalized even for rings with two maximal ideals.
Let r p be a projective rmodule which is not finitely generated. Pdf on semiprojective modules and their endomorphism rings. A wellknown theorem of kaplansky states that every projective module over a local ring is free. In particular, authors considered regularity of endomorphism rings of generalizations principally injective modules 4, 9, 10, 8, 11. Let a1 and a2 be two nonzero rings, and regard them as a1 a2modules via the canonical. Projective module endomorphism ring these keywords were added by machine and not by the authors.
We have discussed the summand intersection property, summand sum property for semiprojective modules. Now let us turn our attention to some examples of projective modules. On rings whose projective modules have the exchange. Endomorphism rings of small pseudo projective modules 605 proposition 4. We further show that one may mod out by the radical, when considering the exchange property for projective modules. The simple modules for a are precisely the simple composition factors of m.
Call a projective module regular if every cyclic submodule is a direct summand. V v, and an endomorphism of a group g is a group homomorphism f. Endomorphism rings of infinitely generated projective modules. Endomorphisms of finetely generated projective modules over a. For this, ais considered as module over the associative multiplication algebra ma and the category. A note on endomorphism rings of semi projective modules article pdf available in mathematical proceedings of the royal irish academy 1122. Constructing endomorphism rings of large finite global dimension.
P p where p is a finitely generated projective amodule a is a com. For example, an endomorphism of a vector space v is a linear map f. Every ring is isomorphic to a subring of the endomorphism ring of an abelian group. Let k be an algebraically closed eld of positive characteristic p, ga nite group, and p a sylow psubgroup of g. Ring extensions and endomorphism rings of a module hirata, kazuhiko and yamashiro, yasukazu, tsukuba journal of. Pdf on sep 1, 1993, jose luis garcia and others published isomorphisms between endomorphism rings of projective modules find, read and cite all the research you need on researchgate. Endomorphism rings of finite global dimension graham j. Introduction a wellknown theorem of kaplansky states that every projective module over a local ring is free. Not every indecomposable module has a local endomorphism ring s, but it happens quite often. Kaplanskys theorem on projective modules wikipedia. It is straightforward that all baer modules are principally projective. Not every indecomposable module has a local endomorphism ring s, but it happens quite often that s is semilocal, or even better. Endomorphism rings and direct sum decompositions in some classes of modules progress in mathematics 1998th edition. Apart from this, we have introduced the idea of mhollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of.
Discovering minimal copresentations of projectives in secure. Let a be a right order in a semisimple ring 2, ma be a finitedimensional torsionless right,4module and mfa be the injective hull of m. In abstract algebra, kaplanskys theorem on projective modules, first proven by irving kaplansky, states that a projective module over a local ring is free. Modules over endomorphism rings this is an extensive synthesis of recent work in the study of endomorphism rings and their modules, bringing together direct sum decompositions of modules, the class number of an algebraic number field, point set topological spaces, and classical noncommutative localization.
Endomorphism rings of projective modules springerlink. Algebraic ktheory of endomorphism rings hongxing chen and changchang xi abstract. Finally, we prove that, any free module f with zero radical is an i 1. On semiprojective modules and their endomorphism rings. We compute the cartan matrix of a and the structure of the projective modules for a. There is a 11 correspondence between numerical semigroups and the rings of formal power series associated to them. R will always have a unit, and every module will be unitary. Projective modules over the endomorphism ring of a biuniform module.
We show that sis regular if and only if every right r module is mgpinjective. Any wgc is isomorphic to a direct summand of the permutation module kg p, where p. Denote by kbrmodand dbrmod the bounded homotopy category and the bounded derived category of r, respectively. Endomorphism rings of small pseudo projective modules. Zelmanowitz has shown that g end ma is a semisimple ring and 5 end ma is a right order in q.
There it is stated that if f is a 2 2matrix with entries in a commutative ring. Finally, we prove that, any free module f with zero radical is an i. More generally, left galois extensions for bialgebroids and their comodule algebras are left d2 8. We investigate in nitely generated projective modules over the endomorphism ring of a biuniform module. Suarez 19 generalized this result to the case for support. However, as we will show in this paper, the theory of infinitely generated projective modules over the endomorphism ring of a biuniform module is interesting and highly nontrivial. Relative singularity categories and gorensteinprojective. Suppose that m is supplemented and a is any submodule of m.
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