Showing papers on "Global dimension published in 1996"
93 citations
77 citations
TL;DR: In this paper, it was shown that if R is a left coherent ring, then the weak global dimension w D(R) = 2 if and only if every (n − 2)th F-cosyzygy of a finitely presented right R-module has a flat envelope with the unique mapping property.
Abstract: We prove that (a) if R is a left coherent ring, then the weak global dimension w D(R) = 2) if and only if every (n – 2)th F–cosyzygy of a finitely presented right R–module has a flat envelope with the unique mapping property; (b) if R is a left coherent and right perfect ring, then the right global dimension rD(R) = 2) if and only if every (n – 2)th P–cosyzygy of a right R–module has a projective envelope with the unique mapping property; (c) if R is a commutative ring, then R is π—coherent (resp. coherent) and the exactness of 0 -> K -> F0 -> F1 with Fo and F1 (finitely) projective and K finitely generated implies the projectivity of K if and only if every finitely generated (resp, finitely presented) R–module has a (finitely) projective envelope with the unique mapping property.
75 citations
TL;DR: In this article, it was shown that the 1-Gorenstein property is inherited by a maximal quotient ring and as a related result, a Noetherian ring of dominant dimension at least 2.
41 citations
TL;DR: In this article, it was shown that the behaviour of these rings R is still quite good, and they have cancellation of small projective modules and their stable range is at most 2.
38 citations
TL;DR: In this article, it was shown that no direct sum of the syzygy modules of k surjects onto a nonzero module of finite projective dimension with nonzero scale.
34 citations
TL;DR: In this article, it was shown that if A→G→Qis a short exact sequence of groups whereGis finitely generated,AandQare abelian,Ais a 2-torsion Krull dimension one-3Q-module via conjugation, then the FPm-Conjecture holds.
29 citations
01 Jul 1996
TL;DR: In this paper, a discussion on the various forms of the Hopkins-Levitzki Theorem for modules and Grothendieck categories and the connection between them can be found.
Abstract: The Hopkins–Levitzki Theorem, discovered independently in 1939 by C. Hopkins and J. Levitzki states that a right Artinian ring with identity is right Noetherian. In the 1970s and 1980s it has been generalized to modules over non-unital rings by Shock[10], to modules satisfying the descending chain condition relative to a heriditary torsion theory by Miller-Teply[7], to Grothendieck categories by Nastasescu [8], and to upper continuous modular lattices by Albu [1]. The importance of the relative Hopkins-Levitzki Theorem in investigating the structure of some relevant classes of modules, including injectives as well as projectives is revealed in [3] and [6], where the main body of both these monographs deals with this topic. A discussion on the various forms of the Hopkins–Levitzki Theorem for modules and Grothendieck categories and the connection between them may be found in [3].
28 citations
TL;DR: In this paper, the Segre product of two quantum planes has been studied in the context of noncommutative algebras, and the main result of this paper is the classification of all embeddings of the segre product into so-called quantum P3's, which are (the Proj of) Artin-Schelter regular algebra with the Hilbert series of a commutative polynomial ring and which map ontoS.
24 citations
TL;DR: In this article, it was shown that if Ao/J(A0) is semisimple, where J(A 0) is the Jacobson radical of A0, then gl. dimA = p.dimA(Ao/J (A0)).
Abstract: Let be a positively graded Noetherian ring. It is proved that if Ao/J(A0) is semisimple, where J(A0) is the Jacobson radical of A0, then gl. dimA = p. dimA(Ao/J(A0)). If A0, is a (not necessarily commutative) local ring with maximal ideal ω , then gl. dimA = p. dimA(A0/ω) = inj. dimA(A0/ω).
13 citations
TL;DR: In this article, a homomorphism u: A → B of Noetherian rings of positive prime characteristic p is associated with the A-algebra structure on A obtained via the r-th power of the Rrobenius endomorphism of A.
Abstract: To a homomorphism u: A → B of Noetherian rings of positive prime characteristic p, we associate the homomorphism given by for each a where ) denotes the A-algebra structure on A obtained via the r-th power of the Rrobenius endomorphism of A. If u is fiat, we show that u is regular if and only if has finite flat dimension if and only if is flat. We also give sufficient conditions for the regularity of a homomorphism u with of finite flat dimension. In the spirit of [A2, Theoreme 23] and [R2, Corollaire 5] we obtain new characterizations for the G-rings of characteristic p. We also state some criteria of Noetherianity for the rings A p ⊗ A B (A B Noetherian rings) and (A fieldB Noetherian ring).
TL;DR: In this article, a generalization of quasi-hereditary rings called QH-1 rings, called serial rings, was introduced and a bound on the global dimension of these rings was given.
Abstract: We give a generalization of quasi-hereditary rings called QH-1 rings. We show that a left QH-1 ring R is quasi hereditary if and only if R has finite global dimension if and only if R has Cartan determinant +1. We then give a bound on the global dimension of the quasi-hereditary serial rings, and charac terize the QH-1 serial rings of infinite global dimension in termi of the admissible sequence.
01 Jan 1996
TL;DR: In this paper, the problem of determining the integer global dimension where the necessary number of coordinates to unfold observed orbits from self overlaps arising from projection of the attractor to a lower dimensional space is addressed.
Abstract: We wish to determine the integer global dimension where we have the necessary number of coordinates to unfold observed orbits from self overlaps arising from projection of the attractor to a lower dimensional space. For this we go into the data set and ask when the unwanted overlaps occur. The lowest dimension which unfolds the attractor so that none of these overlaps remains is called the embedding dimension d E . d E is an integer. If we measure two quantities s A (n) and s B (n) from the same system, there is no guarantee that the global dimension d E for from each of these is the same. Each measurement along with its timelags provides a different nonlinear combination of the original dynamical variables and can provide a different global nonlinear mapping of the true space x(n) into a reconstructed space of dimension d E where smoothness and uniqueness of the trajectories is preserved. Recall that d E is a global dimension and may well be different from the local dimension of the underlying dynamics.
01 Aug 1996
TL;DR: In this article it was shown that if every finitely generated submodule of E embeds in a finitely presented module of projective dimension < 1, then every finitley generated right S/J-module X is canonically isomorphic to HomR(E, X $S E).
Abstract: Let R be a ring, E = E(RR) its injective envelope, S = End(ER) and J the Jacobson radical of S. It is shown that if every finitely generated submodule of E embeds in a finitely presented module of projective dimension < 1, then every finitley generated right S/J-module X is canonically isomorphic to HomR(E, X $S E). This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, E/JE is completely pure-injective (a property that holds, for example, when the right pure global dimension of R is < 1 and hence when R is a countable ring), then S is semiperfect and RR is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.
01 Jan 1996
TL;DR: A quasi-hereditary algebra with triangular decomposition CAC-P-C 5 C0 P such that all Verma modules are semisimple over C"P was studied in this paper.
Abstract: Let A be a quasi-hereditary algebra with triangular decomposition CAC-P-C 5 C0 P such that all Verma modules are semisimple over C"P. Then we show: gldim(A) = 2 . gldim(C). Applying this formula to the more special class of twisted double incidence algebras of finite partially ordered sets, we get a proof of a conjecture of Deng and Xi. Another application is to the so-called dual extensions of algebras.
TL;DR: In this article, it was shown that any ring with Krull dimension satisfies the ascending chain condition on semiprime ideals, and that any R-module with a polynomial identity satisfies the same condition on prime submodules.
Abstract: Any ring with Krull dimension satisfies the ascending chain condition on semiprime ideals. This result does not hold more generally for modules. In particular if Ris the first Weyl algebra over a field of characteristic 0 then there are Artinian R-modules which do not satisfy the ascending chain condition on prime submodules. However, if Ris a ring which satisfies a polynomial identity then any R-module with Krull dimension satisfies the ascending chain condition on prime submodules, and, if Ris left Noethe-rian, also the ascending chain condition on semiprime submodules.
TL;DR: In this article, abstract localization and graded versions of the Artin-Rees property are applied to construct structure sheaves over the projective spectrum Proj(R) of a graded fully bounded noetherian ring R.
Abstract: In this note, we show how abstract localization and graded versions of the Artin-Rees property may be applied to construct structure sheaves over the projective spectrum Proj(R) of a graded fully bounded noetherian ring R.
TL;DR: Gelfand-Kirillov dimension (GK) has proved to be a useful invariant for algebras over fields as mentioned in this paper, and the notion of GK has been generalized to algebra over commutative Noetherian rings by replacing vector space dimension with reduced rank.
Abstract: Gelfand-Kirillov dimension (GK) has proved to be a useful invariant for algebras over fields. In this paper we generalize the notion of GK to algebras over commutative Noetherian rings by replacing vector space dimension with reduced rank. It turns out that most results about GK have analogues for the new GK.
Journal Article•
TL;DR: In this paper, it was shown that the Noetherian semi-local property of the underlying ring enables us to develop a setisfactory concep of the theory of reduction of ideals in a commutative noetherian ring.
Abstract: The purpose of this paper is to show that the Noetherian semi-local property of the underlying ring enables us to develope a setisfactory concep of the theory of reduction of ideals in a commutative Noetherian ring.
TL;DR: In this article, the global dimension of certain sub-rings of the matrix ring M n(R), where R is an arbitrary ring with 1≠,0, was bound, in the case when R is a commutative Noethenan domain.
Abstract: In this paper we shall bound the global dimension of certain subrings of the matrix ring M n(R), where R is an arbitrary ring with 1≠,0. We define a class of rings which satisfy a triangular decomposition. This includes triangular tiled matrix rings. For such rings, we provide a bound which, in the case when R is a commutative Noethenan domain, gives a wider class of rings satisfying a conjecture of Tarsy.
TL;DR: In this article, a generalization of Gelfand Kirillov dimension GGK was proposed for affine Noetherian PI algebras and it was shown that GGK at most one is PI for a large class of commutative base rings including the ring of integers, Z.
Abstract: Using a growth function,GK defined for algebras over integral domains, we construct a generalization of Gelfand Kirillov dimensionGGK. GGK coincides with the classical no-tion of GK for algebras over a field, but is defined for algebras over arbitrary commutative rings. It is proved that GGK exceeds the Krull dimension for affine Noetherian PI algebras. The main result is that algebras of GGK at most one are PI for a large class of commutative Noetherian base rings including the ring of integers, Z. This extends the well-known result of Small, Stafford, and Warfield found in [11].
TL;DR: In this paper, the problem of the Krull dimension of rings of skew polynomials is studied for Weyl algebras, a ring of differential operators, and rings of Laurent skew polytopes.
Abstract: The Krull dimension of rings of skew polynomials is studied. Earlier the problem of Krull dimension was investigated only for some particular cases, namely, for Weyl algebras [2], a ring of differential operators [7,8], as well as for rings of Laurent skew polynomials [9–10].
01 Jan 1996
TL;DR: In this paper, the weak algorithm for polynomial and power series rings is described, which provides a counterpart to the Euclidean algorithm and forms a natural tool for the study of polynomials in several noncommuting indeterminates.
Abstract: This chapter discusses polynomial and power series rings. It describes the weak algorithm, which provides a counterpart to the Euclidean algorithm, and it forms a natural tool for the study of polynomials in several noncommuting indeterminates. As in a Euclidean domain, every ideal is principal, so the (one-sided) ideals in a ring with weak algorithm are free, as modules over the ring, and this leads to firs and semifirs. Coproducts, a natural extension of free algebras, are also summarized. For any ring, R the polynomial ring R [ x ] is a familiar construction, obtained by adjoining to R an element x subject to the rule: ax = xa , for all a ∈ R . Moreover, every element of R [ x ] can be uniquely expressed as a polynomial in x with coefficients from R . In homological algebra, the global dimension of a ring forms a means of classification, and the hereditary rings—that is, the rings of global dimension 1—are simplest after the familiar case of global dimension 0 (the semisimple rings). A second mode of classification is based on the form taken by projective modules. The simplest class is formed by the projective-free rings, in which every finitely generated projective right module is free, of unique rank.