1,720,961 research outputs found
The Krull-Schmidt Theorem in the case two
No full textWe study what happens if, in the Krull-Schmidt Theorem, instead of considering modules whose endomorphism rings have one maximal ideal, we consider modules whose endomorphism rings have two maximal ideals. If a ring has exactly two maximal right ideals, then the two maximal right ideals are necessarily two-sided. We call such a ring of type 2. The behavior of direct sums of finitely many modules whose endomorphism rings have type 2 is completely described by a graph whose connected components are either complete graphs or complete bipartite graphs. The vertices of the graphs are ideals in a suitable full subcategory of Mod-R. The edges are isomorphism classes of modules. The complete bipartite graphs give rise to a behavior described by a Weak Krull-Schmidt Theorem. Such a behavior had been previously studied for the classes of uniserial modules, biuniform modules, cyclically presented modules over a local ring, kernels of morphisms between indecomposable injective modules, and couniformly presented modules. All these modules have endomorphism rings that are either local or of type 2. Here we present a general theory that includes all these cases
Monogeny dimension relative to a fixed uniform module
No full textFor a module A and a uniform module U, we consider the invariant m-dim_U(A):=sup{i\in \N_0 | there exist morphisms f:U_i→A and g:A→U_i with gf a monomorphism}. This invariant turns out to have the following properties: (1) m-dim_U respects direct sums; (2) if U and V are uniform and [U]_m=[V]_m, then m-dim_U=m-dim_V; and (3) if two modules A and B have finite Goldie dimension and [A]_m=[B]_m, then m-dim_U(A)=m-dim_U(B) for every uniform module U. In particular, when A has finite Goldie dimension and is a direct summand of a serial module, the values m-dim_U(A) completely determine the monogeny class of the module A. We give a complete description of the monoid of all isomorphism classes of serial modules of finite Goldie dimension over a fixed ring R
Covering classes and uniserial modules
No full textWe apply minimal weakly generating sets to study the existence of Add(UR)-covers for a uniserial module UR. If UR is a uniserial right module over a ring R, then S:=End(UR) has at most two maximal (right, left, two-sided) ideals: one is the set I of all endomorphisms that are not injective, and the other is the set K of all endomorphisms of UR that are not surjective. We prove that if UR is either finitely generated, or artinian, or I⊂K, then the class Add(UR) is covering if and only if it is closed under direct limit. Moreover, we study endomorphism rings of artinian uniserial modules giving several examples
Representations of the category of serial modules of finite Goldie dimension
No full textWe study the category SUsr of all serial right modules of finite Goldie dimension over a fixed ring R. This category has natural valuations m-dimU and e-dimU for every uniserial right R-module U, because the number m-dimU(A) (e-dimU(A)) of modules in the same monogeny (epigeny) class as U in any indecomposable direct-sum decomposition of an object A of SUsr is uniquely determined by A. These valuations m-dimU, e-dimU: V (SUsr) → N0 are not essential valuations in general, so that the quotient categories of SUsr corresponding to the valuations m-dimU, e-dimU are Krull-Schmidt categories but not IBN categories. With these valuations, we obtain a natural representation SUsr → <i∈I Ai that is an isomorphism reflecting and direct-summand reflecting additive functor. Here the categories Ai are quotient Krull-Schmidt categories of SUsr. If, instead of considering all the valuations m-dimU, e-dimU of SUsr that come from the monogeny classes and the epigeny classes of all uniserial right R-modules, we consider only those that are essential valuations, then we get a divisor theory of the monoid V (SUsr) and, correspondingly, a representation SUsr → <i∈J Ai in which the categories Ai, i ∈ J, are IBN categories. © de Gruyter 2008
Pure projective tilting modules
Full text linkLet TR be a 1-tilting module with tilting torsion pair (Gen T , F ) in Mod-R. The following conditions are proved to be equiv- alent: (1) T is pure projective; (2) Gen T is a definable subcategory of Mod-R with enough pure projectives; (3) both classes GenT and F are finitely axiomatizable; and (4) the heart of the corresponding HRS t-structure (in the derived category Db(Mod-R)) is Grothendieck. This article explores in this context the question raised by Saor ́ın if the Grothendieck condition on the heart of an HRS t-structure implies that it is equivalent to a module category. This amounts to asking if T is tilting equivalent to a finitely presented module. This is re- solved in the positive for a Krull-Schmidt ring, and for a commutative ring, a positive answer follows from a proof that every pure projective 1-tilting module is projective. However, a general criterion is found that yields a negative answer to Saor ́ın’s Question and this criterion is satisfied by the universal enveloping algebra of a semisimple Lie algebra, a left and right noetherian domain
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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