Hereditary Rings Research Articles

Partial Skew Polynomial Rings Over Semisimple Artinian Rings

Let R be a semisimple Artinian ring with a partial action α of ℤ on R, and let R[x; α] be the partial skew polynomial ring. By the classification of the set E of all minimal central idempotents in R into three different types, a complete description of the prime radical of R[x; α] is given. Moreover, it is shown that any nonzero prime ideal of R[x; α] is maximal and is either principal or idempotent. In the case where α is of finite type, it is shown that R[x; α] is a semiprime hereditary ring.

On the endofiniteness of a key module over pure semisimple rings

Let R R be a left pure semisimple ring such that there are no non-zero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R R -mod, and let W W be the left key R R -module; i.e., W W is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left R R -modules. We show that if the module W W is endofinite, then R R is a ring of finite representation type. This settles a question considered in [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007), 361-376] for hereditary rings.

Pure Hereditary Rings

In this article, the notion of pure hereditary rings is introduced as a nontrivial generalization of the notion of both pure semisimple rings and of hereditary rings. Some properties and examples of pure hereditary rings are given. For a positive integer n, let 𝒫 n be the class of all left R-modules with projective dimension ≤ n. It is shown that over a left pure hereditary ring is a perfect and hereditary cotorsion pair, which gives a partial positive answer to a question of Göbel and Trlifaj. The modules in the class are also studied. Special attention is paid to the case n = 1.

On direct sums of Baer modules

The notion of Baer modules was defined recently. Since a direct sum of Baer modules is not a Baer module in general, an open question is to find necessary and sufficient conditions for such a direct sum to be Baer. In this paper we study rings for which every free module is Baer. It is shown that this is precisely the class of semiprimary hereditary rings. We also prove that every finite rank free R-module is Baer if and only if R is right semihereditary, left Π-coherent. Necessary and sufficient conditions for finite direct sums of copies of a Baer module to be Baer are obtained, for the case when M is retractable. An example of a module M is exhibited for which M n is Baer but M n + 1 is not Baer. Other results on direct sums of Baer modules to be Baer under some additional conditions are obtained. Some applications are also included.

Preinjective modules over pure semisimple rings

For a left pure semisimple ring R , it is shown that the local duality establishes a bijection between the preinjective left R -modules and the preprojective right R -modules, and any preinjective left R -module is the source of a left almost split morphism. Moreover, if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R -mod, the direct sum of all non-preinjective indecomposable direct summands of products of preinjective left R -modules is a finitely generated product-complete module. This generalizes a recent theorem of Angeleri Hügel [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007) 361–376] for hereditary rings.

Finitely Presented Modules Over Semihereditary Rings

We prove that each almost local-global semihereditary ring R has the stacked bases property and is almost Bézout. More precisely, if M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annihilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a projective module which is isomorphic to a direct sum of finitely generated ideals. These ideals allow us to define a finitely generated ideal whose isomorphism class is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too. It follows that each semihereditary ring of Krull-dimension one or of finite character, in particular each hereditary ring, has the stacked base property. These results were already proved for Prüfer domains by Brewer, Katz, Klinger, Levy, and Ullery. It is also shown that every semihereditary Bézout ring of countable character is an elementary divisor ring.

Global dimension and left derived functors of Hom

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R) ⩽ n (n ⩾ 2) if and only if the gl right Proj-dim MR ⩽ n−2 if and only if Extn−1(N,M) = 0 for all right R-modules N and M if and only if every (n−2)th Projcosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R) ⩽ n (n ⩾ 1) if and only if every (n−1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Proj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R) ⩽ 2 are characterized.

Some External Characterizations of SV-Rings and Hereditary Rings

We introduce a new class of rings which we callAI-rings. By applying this property to endomorphism rings, we give characterizations of semi-ArtinianV-rings and hereditary rings.

On Serial Noetherian Rings

It is well known that every serial Noetherian ring satisfies the restricted minimum condition. In particular, following Warfield (1975), such a ring is a direct sum of an Artinian ring and hereditary prime rings. The aim of this note is to show that every serial ring having the restricted minimum condition is Noetherian.

On the Goldie dimension of rings and modules

We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generating sets of their quasi-injective hulls. Several consequences are deduced. In particular, it is shown that every finitely generated hereditary module with countably generated quasi-injective hull is noetherian. It is also shown that every right hereditary ring with finitely generated injective hull is right artinian, thus answering a long standing open question posed by Dung, Gómez Pardo and Wisbauer.

Hereditary and Semiperfect Distributive Rings

A ring R is called right distributive if its lattice of right ideals is distributive. In this paper, we investigate distributive rings. We prove that if a ring R is right hereditary, then R is right distributive if and only if R is weakly right duo. We also prove that right semiperfect right distributive rings are right quasi-continuous. Finally, it is proved that right perfect distributive rings are quasi-Frobenius. In addition, we add examples to the situations that occur naturally in the process of this paper.

Weakly Stable Modules

ABSTRACT A new notion which is called weakly stable module is introduced in this article. It is a nontrivial generalization of the modules with endomorphism rings having stable range one. We deduce that weakly stable projective modules have the cancellation property, and so any commutative hereditary ring has the cancellation property, i.e., if R is a commutative hereditary ring, then for any R-modules B and C, R ⊕ B ≅ R ⊕ C implies B ≅ C.

The Structure of the Reduced Product of Preinjective Modules Over a Hereditary Right Pure Semisimple Ring

ABSTRACT Let G be a division ring and F ⊂ G a fixed subdivision ring of G such that dim F G = ∞, dim G F = 2 and the F-G-bimodule F G G has the infinite dimension type d−∞( F G G ) = (…, 2, 2…,2, 2, 1, ∞) in the sense of Simson (19962000), see Section 2. In particular, the hereditary ring is a counter-example to the pure semisimplicity conjecture. Let be a complete set of representatives of pairwise non-isomorphic preinjective right R G -modules. We prove that the reduced product is a direct sum , where , , s 0 = s 1 = (card G)ℵ0 , and means the direct sum of s j copies of the right ideal P j of R G . We also show that is isomorphic to ∏ℵ0 P 1. Communicated by J. Gomaz Pardo.

Triangulated Categories and the Ziegler Spectrum

The relationship between the Ziegler spectrum of (the category of modules over) a ring and the Ziegler spectrum of its derived category is investigated. Over von Neumann regular rings and hereditary rings the spectrum of the derived category is a disjoint union of copies of the spectrum of the ring but in general there are further indecomposable pure-injective objects of the derived category.

Purities and Pure Injective Modules over Serial, Right Noetherian Rings

It is proved that any submodule of the direct sum of a family of finitely generated right modules over a serial, right Noetherian ring is also a direct sum of finitely generated modules. With the help of reduction to serial, right hereditary rings, we obtained a new description of all indecomposable pure injective noninjective right modules over serial, right Noetherian rings. Bibliography: 22 titles.

A new characterisation of hereditary pi rings

In this paper we prove a new characterisation of hereditary PI rings, namely we show that a Noetherian, but not Artinian, PI ringR that is an order in an Artinian ring splits into a direct sum of an Artinian ring of finite representation type and hereditary semiprime rings if and only if all its proper Artinian factor rings are of finite representation type. We also show, through examples, that the above characterisation does not hold for some more general settings.

Triangular matrix representations of ring extensions

In this paper we investigate the class of piecewise prime, PWP, rings which properly includes all piecewise domains (hence all right hereditary rings which are semiprimary or right Noetherian). For a PWP ring we determine a large class of ring extensions which have a generalized triangular matrix representation for which the diagonal rings are prime.

On strongly almost hereditary rings

On strongly almost hereditary rings

DIRECT SUM DECOMPOSITION OF THE PRODUCT OF PREINJECTIVE MODULES OVER RIGHT PURE SEMISIMPLE HEREDITARY RINGS

ABSTRACT In his recent work, [1] and [2], on the pure semisimplicity conjecture Simson raised two problems about the structure of the direct sum decomposition of the direct product modulo the direct sum of indecomposable preinjective modules over right pure semisimple hereditary rings. The main goal of this paper is the proof of a theorem that resolves one of these problems and provides a partial answer to the other.

Quasi-Frobenius Rings and Nakayama Permutations of Semiperfect Rings

We say that $${\mathcal{A}}$$ is a ring with duality for simple modules, or simply a DSM-ring, if, for every simple right (left) $${\mathcal{A}}$$ -module U, the dual module U* is a simple left (right) $${\mathcal{A}}$$ -module. We prove that a semiperfect ring is a DSM-ring if and only if it admits a Nakayama permutation. We introduce the notion of a monomial ideal of a semiperfect ring and study the structure of hereditary semiperfect rings with monomial ideals. We consider perfect rings with monomial socles.