Minimal Injective Cogenerator Research Articles

Dualidad de Matlis generalizado, sobre un anillo conmutativo Noetheriano

Let R be a Noetherian commutative ring and let E be the minimal injective cogenerator of the category of R-modules. Belshoff, Enochs and García Rozas are three people who introduced in [6] the so-called I-reflexive Matlis modules where I is an ideal. In this context we present in the second section a brief introduction of the reflective modules and some of their properties, and in the third and last section we give the classification of reflective modules with respect to a minimal cogenerator E. A node M is said to be reflexive with respect to E if the map of M in Hom(HomR(M, E), E) is an isomorphism and thus we establish the classification of the R-modules M, which are reflexive with respect to E if and only if M has a finitely generated submodule S such that the quotients: M/S and R/anul(M) is complete Artinian and semilocal respectively.

Injective modules over the Jacobson algebra

AbstractFor a field K, let $\mathcal {R}$ denote the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle $ . We give an explicit construction of the injective envelope of each of the (infinitely many) simple left $\mathcal {R}$ -modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for $\mathcal {R}$ . Our approach involves realizing $\mathcal {R}$ up to isomorphism as the Leavitt path K-algebra of an appropriate graph $\mathcal {T}$ , which thereby allows us to utilize important machinery developed for that class of algebras.

Almost Gentle Algebras and their Trivial Extensions

AbstractIn this paper we define almost gentle algebras, which are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and, as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one is the trivial extension of an almost gentle algebra. We show that a hypergraph is associated with every almost gentle algebra A, and that this hypergraph induces the Brauer configuration of the trivial extension of A. Among other things, this gives a combinatorial criterion to decide when two almost gentle algebras have isomorphic trivial extensions.

A change of rings result for Matlis reflexivity

Let $R$ be a commutative Noetherian ring and $E$ the minimal injective cogenerator of the category of $R$-modules. An $R$-module $M$ is (Matlis) reflexive if the natural evaluation map $M \to \operatorname{Hom}_R(\operatorname{Hom}_R(M,E),E)$ is an isomorphism. We prove that if $S$ is a multiplicatively closed subset of $R$ and $M$ is a reflexive $R$-module, then $M$ is a reflexive $R_S$-module. The converse holds when $S$ is the complement of the union of finitely many minimal primes of $R$, but fails in general.

Trivial extensions of gentle algebras and Brauer graph algebras

We show that two well-studied classes of tame algebras coincide: namely, the class of symmetric special biserial algebras coincides with the class of Brauer graph algebras. We then explore the connection between gentle algebras and symmetric special biserial algebras by explicitly determining the trivial extension of a gentle algebra by its minimal injective co-generator. This is a symmetric special biserial algebra and hence a Brauer graph algebra of which we explicitly give the Brauer graph. We further show that a Brauer graph algebra gives rise, via admissible cuts, to many gentle algebras and that the trivial extension of a gentle algebra obtained via an admissible cut is the original Brauer graph algebra.As a consequence we prove that the trivial extension of a Jacobian algebra of an ideal triangulation of a Riemann surface with marked points in the boundary is isomorphic to the Brauer graph algebra with Brauer graph given by the arcs of the triangulation.

Regular semiartinian rings

We study the structure of rings over which every right module is an essential extension of a semisimple module by an injective one. A ring R is called a right max-ring if every nonzero right R-module has a maximal submodule. We describe normal regular semiartinian rings whose endomorphism ring of the minimal injective cogenerator is a max-ring.

Isomorphic trivial extensions of finite dimensional algebras

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ . Let T ( Λ ) = Λ ⋉ D ( Λ ) be the trivial extension of Λ by its minimal injective cogenerator D ( Λ ) . We characterize, in terms of quivers and relations, the algebras Λ ′ such that T ( Λ ) ≃ T ( Λ ′ ) .

GENERALIZED MATLIS DUALITY AND LINEAR COMPACTNESS

ABSTRACT Let R be a commutative ring. We show that a finitely generated R-module M is reflexive (equivalently, linearly compact) if and only if R/annR(M) is reflexive (equivalently, linearly compact). If R is a quasi-noetherian ring, we prove that an R-module M is reflexive if and only if R/annR(M) is linearly compact and M has a finitely generated submodule N such that M/N is finitely cogenerated. These properly generalize results of Belshoff, Enochs and Garcia Rozas. A ring extension S of R is considered and we show that under some minor condition an S-module is reflexive if and only if it is reflexive as an R-module. Throughout this paper, rings are commutative with identity and modules are unitary. The terminology and notations not defined here can be found in Anderson and Fuller [1]. Let R be a ring and be the minimal injective cogenerator of the category of R-modules, where is an irredundant set of representatives of the simple R-modules. If M is an R-module we let . Following Belshoff, Enochs and Garcia Rozas [2], we call M a (Matlis) reflexive R-module if the evaluation map is an isomorphism. Note that eM is always injective, since U is a cogenerator of R-modules. We recall that an R-module M is linearly compact if every finitely solvable family (where each ) is solvable. For the convenience of the reader we record some basic properties of linearly compact modules and reflexive modules as follows (see [[6], Sec. 3] and [2]): (1) Every artinian module is linearly compact; (2) an infinite direct sum of non-zero modules is not linearly compact (reflexive); (3) If N is a submodule of a module M, then M is linearly compact (reflexive) if and only if both N and M/N are linearly compact (reflexive); consequently, a finite direct sum of modules is linearly compact (reflexive) if and only if each summand is linearly compact (reflexive); and (4) If M is an R-module and IM=0 for an ideal I of R, then M is linearly compact (reflexive) as an R-module if and only if it is linearly compact (reflexive) as an (R/I)-module. This paper consists of two sections. In Sec. 1, we show that the following are equivalent for a finitely generated R-module M: (1) M is reflexive; (2) M is linearly compact; (3) is linearly compact; (4) is reflexive. If R is a quasi-noetherian ring, i.e., the ideal annR(M) is finitely generated for each finitely cogenerated reflexive R-module M, we show that an R-module M is reflexive if and only if R/annR(M) is linearly compact and M has a finitely generated submodule N such that M/N is finitely cogenerated. In Sec. 2, we consider a ring extension S of R. We prove that is the minimal injective cogenerator of S-modules and an S-module M is reflexive as an S-module if and only if M is reflexive as an R-module, provided that one of the following three conditions is satisfied: (1) S is finitely generated as an R-module, (2) S is linearly compact as an R-module, or (3) S=RQ is a split extension with Q nilpotent.

Presentations of Trivial Extensions of Finite Dimensional Algebras and a Theorem of Sheila Brenner

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. We describe the ordinary quiver and relations for T(Λ)=Λ⋉D(Λ), the trivial extension of Λ by its minimal injective cogenerator D(Λ), and also for the repetitive algebra [formula] of Λ. Associated with this description we give an application of a theorem of Sheila Brenner.

Generalized Matlis duality

Let R R be a commutative noetherian ring and let E E be the minimal injective cogenerator of the category of R R -modules. A module M M is said to be reflexive with respect to E E if the natural evaluation map from M M to Hom R ⁡ ( Hom R ⁡ ( M , E ) , E ) \operatorname {Hom}_R( \operatorname {Hom}_R(M,E), E) is an isomorphism. We give a classification of modules which are reflexive with respect to E E . A module M M is reflexive with respect to E E if and only if M M has a finitely generated submodule S S such that M / S M/S is artinian and R / ann ⁡ ( M ) R/\operatorname {ann}(M) is a complete semi-local ring.

A note on self-dual modules and Dedekind rings

It is shown that a commutative noetherian ring is a finite direct sum of Dedekind rings and artinian uniserial rings if and only if every module of finite length is selfdual. A module of finite length is said to be selfdual if it is isomorphic to its dual with respect to the minimal injective cogenerator.

Injective envelopes and flat covers of modules over a commutative ring

Let R be a commutative semilocal ring and U be the minimal injective cogenerator in the category of R-modules. If R is almost noetherian, i.e., each non-minimal prime ideal of R is finitely generated, we prove that the following three statements are equivalent: (1) every U-reflexive R-module has a U-reflexive injective envelope; (2) every U-reflexive R-module has a U-reflexive flat cover; and (3) R is linearly compact and the Krull dimension of R is less than or equal to 1. Our examples show that this result is a non-trivial generalization of a recent theorem of Belshoff and Xu, and that the condition R being almost noetherian is essential.

On the tameness of trivial extension algebras

For a finite dimensional algebra A over an algebraically closed field, let T(A) denote the trivial extension of A by its minimal injective cogenerator bimodule. We prove that, if $T_A$ is a tilting module and $B=End T_A$, then T(A) is tame if and only if

Auslander-Reiten triangles in derived categories of finite-dimensional algebras

Let A A be a finite-dimensional algebra. The category b m o d A bmod A of finitely generated left A A -modules canonically embeds into the derived category D b ( A ) {D^b}\left ( A \right ) of bounded complexes over b m o d A bmod A and the stable category mod _ Z T ( A ) {\underline {\bmod } ^\mathbb {Z}}T\left ( A \right ) of Z \mathbb {Z} -graded modules over the trivial extension algebra of A A by the minimal injective cogenerator. This embedding can be extended to a full and faithful functor from D b ( A ) {D^b}\left ( A \right ) to mod _ Z T ( A ) \underline {\bmod }^{\mathbb {Z}}T\left ( A \right ) . Using the concept of Auslander-Reiten triangles it is shown that both categories are equivalent only if A A has finite global dimension.

Domestic trivial extensions of simply connected algebras

Let A be a finite-dimensional,basicand connected algebra (associative,with 1) over an algebraicallyclosedfield.Itis called simply connected ititis triangular and, for any presentation of A as a bound quiver algebra, the fundamental group of itsbound quiver is trivial. Let T(A) denote thetrivialextension of A by its minimal injective cogenerator. We show that,if A is simply con- nected, then the following conditions are equivalent: (i)T{A) is representation-infiniteand domestic, (ii)T(A) is 2-parametric, (iii) there exists a representation-infinitetiltedalgebra B of Euclidean type Dn or Ep such that T(A)2^T(B), (iv)A is an iterated tilted algebra of type Dn or Ev.

On modules with trivial self-extensions

Throughout this note, all rings will be self-basic connected artin algebras over a fixed commutative local artin ring k and all modules will be finitely generated. Homomorphisms will be written on the opposite side of the scalars. By mod-A (resp. A-mod), we denote the category of all right (resp. left) A-modules for an artin algebra A. The ordinary duality functor is denoted by D: mod-A 2 A-mod, i.e., D = Hom,(?, f) with the minimal injective cogenerator I over k. We are interested in the bimodules BTA with the following properties:

Representation-finite trivial extension algebras

Let A be a finite-dimensional basic connected associative algebra over an algebraically closed field, and T(A)=A ⋉ DA its trivial extension by its minimal injective cogenerator. We prove that T( A) is representation-finite of Cartan class Δ if and only if A is an iterated tilted algebra of Dynkin class Δ. The proof also yields a construction procedure for iterated tilted algebras of Dynkin type.

Rings of Equivalent Dominant and Codominant Dimensions

Continuing an earlier examination of the codominant dimension of rings and modules, a categorical characterization is given for rings of equivalent dominant and codominant dimensions. Specifically, the question is reduced to when ${}_RR$ and the left minimal injective cogenerator ${}_R\mathcal {U}$ can be used as test modules respectively for the dominant dimension of the projective modules and the codominant dimension of the injective modules. These conditions are in turn characterized by when the injective projective modules are $\mathbf {\Sigma }$-injective. Also, a new and shortened version is given for the proof that the codominant dimension of the injective modules is equal to the dominant dimension of the projective modules.

Rings of equivalent dominant and codominant dimensions

Continuing an earlier examination of the codominant dimension of rings and modules, a categorical characterization is given for rings of equivalent dominant and codominant dimensions. Specifically, the question is reduced to when R R and the left minimal injective cogenerator R can be used as test modules respectively for the dominant dimension of the projective modules and the codominant dimension of the injective modules. These conditions are in turn characterized by when the injective projective modules are Z-injective. Also, a new and shortened version is given for the proof that the codominant dimension of the injective modules is equal to the dominant dimension of the projective modules. Throughout, R is an associative ring with unity and all modules considered are unital. We denote a minimal injective cogenerator for R)R by R such being isomorphic to an injective hull of a direct sum of a complete representative class of simple left R-modules. For a left perfect ring R and a left R-module RM we define the codominant dimension of RM, denoted codom dim RM, to be greater than or equal to n if for a minimal projective resolution . P* P M 0, Pi is injective for i = 1, 29, . . ., n. The left codominant dimesnion of R is defined to be codom dim RU. The dominant dimension of RM is defined in a dual manner, and the left dominant dimension of R is dom dimRR. Further, for R left perfect, we define codom dimR9 to be infIcodom dim RE: RE injectivel. Similarly, dom dim Ris given by inf dom dimRP: RP projectivel. Finally, we say that RE (nRP) is a test module for RI (RT) provided codom dim RE= codom dimRI (dom dimR P = dom dimRY). In [2] we showed that dom dim R 9= codom dim RI. As an application it can Presented to the Society, January 25, 1973; received by the editors April 16, 1973. AMS (MOS) subject classifications (1970). Primary 16A60; Secondary 16A50, 16A52.