Commutative Local Ring Research Articles

Cofiniteness of generalized local cohomology modules over local rings

Let [Formula: see text] be a local commutative Noetherian ring and [Formula: see text] an ideal of [Formula: see text] Assume that [Formula: see text] is a finitely generated [Formula: see text]-module of finite projective dimension [Formula: see text] and [Formula: see text] is a finitely generated [Formula: see text]-module of dimension [Formula: see text] We prove that [Formula: see text] is Artinian. Moreover, if [Formula: see text] is finitely generated, then [Formula: see text] is an [Formula: see text]-cofinite [Formula: see text]-module.

Locally cohomologically complete intersection ideals and cofiniteness

Let [Formula: see text] be a unitary commutative local Noetherian ring and [Formula: see text] be an ideal. The aim of this paper is twofold: In the first part of this paper, we consider locally cohomologically complete intersection ideals of pure height [Formula: see text]. Next, we deal with cofinite modules. In particular, we investigate conditions for cofiniteness of [Formula: see text] and conditions under which, the modules [Formula: see text] are of finite length for all [Formula: see text] and all finitely generated [Formula: see text]-module [Formula: see text].

Rings of finite Morley rank without the canonical base property

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over an algebraically closed field of characteristic [Formula: see text] give rise to triangular rings without the CBP. This also applies to Baudisch’s [Formula: see text]-step nilpotent Lie algebras, which yields the existence of a [Formula: see text]-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP.

Number of cliques of Paley-type graphs over finite commutative local rings

Number of cliques of Paley-type graphs over finite commutative local rings

Von Neumann local matrices

We use our recent results on von Neumann regular matrices, strongly regular matrices and matrices having a non-zero outer inverse to derive applications to some generalizations of these concepts, called von Neumann local, strongly von Neumann local and outer von Neumann local matrices. Among other properties, we show that the $t^{\rm th}$ compound matrix of every matrix of determinantal rank $t$ over a commutative local ring is strongly von Neumann local, and every matrix over an arbitrary semiperfect ring is outer von Neumann local.

Commutative rings whose proper ideals decompose into local modules

For a commutative local ring R, we know that every ideal is a direct sum of local modules if and only if the unique maximal ideal of R is of the form , where and are principal ideal rings and for all γ’s. This motivates us to determine a commutative ring, not necessarily local, in which every (proper) ideal is a direct sum of local modules. We prove that such a ring is exactly a finite direct product of local rings R with the unique maximal ideal in the above mentioned form. Moreover, we characterize more precisely certain commutative rings such as Noetherian rings, reduced rings, Rickart rings, semi-Artinian rings and self-injective rings, in which every (proper) ideal is a direct sum of local modules.

EXACT SUBCATEGORIES, SUBFUNCTORS OF , AND SOME APPLICATIONS

Abstract Let $({\cal{A}},{\cal{E}})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of ${\operatorname{Ext}}_{\cal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in detail and find a range of applications from detecting regularity to understanding Ulrich modules.

Waring numbers over finite commutative local rings

In this paper we study Waring numbers gR(k) for (R,m) a finite commutative local ring with identity and k∈N with (k,|R|)=1. We first relate the Waring number gR(k) with the diameter of the Cayley graphs GR(k)=Cay(R,UR(k)) and WR(k)=Cay(R,SR(k)) with UR(k)={xk:x∈R⁎} and SR(k)={xk:x∈R×}, distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph GR(k) can be obtained by blowing-up the vertices of GFq(k) a number |m| of times, with independence sets the cosets of m, where q is the size of the residue field R/m. Then, by using the above blowing-up, we reduce the study of the Waring number gR(k) over the local ring R to the computation of the Waring number g(k,q) over the finite residue field R/m≃Fq. In this way, using known results for Waring numbers over finite fields, we obtain several explicit results for Waring numbers over finite commutative local rings with identity.

Cellular Noetherian algebras with finite global dimension are split quasi-hereditary

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers.

Two families of few-weight codes over a finite chain ring

Linear codes over finite chain rings meeting the Griesmer bound are particularly interesting since they are closely related to minihyper in projective Hjelmslev spaces. In this paper, motivated by the nice article (Li et al., 2018 [23]), two classes of linear codes with few non-zero Hamming weights over the finite chain ring R=Fq+uFq are constructed, and one of them meets the Griesmer bound over finite commutative local rings. By a Gray map, we obtain that the Gray images of the constructed codes over the ring R are linear codes with two weights over the finite field Fq with q elements, and these codes are optimal with respect to the well-known Griesmer bound under certain conditions. Our results give an affirmative answer to two (related) open questions raised in the article mentioned above. Remarkably, some of the constructed linear codes over R presented in this paper correspond to minihypers in projective Hjelmslev spaces over finite chain rings. Besides, their Gray images can be used to construct secret sharing schemes with nice access structures and other applications thanks to the minimality property that possesses our constructed codes.

On the set of all generalized Drazin invertible elements in a ring

Berkani and Sarihr [Studia Math. (2001) 148: 251-257] showed that the set of all Drazin invertible elements in an algebra over a filed is a regularity in the sense of Kordula and M?ller [Studia Math. (1996) 119: 109-128]. In this paper, the above result is extended to the case of a ring. Counterexamples are provided to show that the set of all generalized Drazin invertible elements in a ring need not be a regularity in general. We determine when the set of all generalized Drazin invertible matrices in the 2 ? 2 full matrix ring over a commutative local ring is a regularity. We also give a sufficient condition for the set of all generalized Drazin invertible elements in a ring to be a regularity.

Strongly regular matrices revisited

We prove a necessary condition and a sufficient condition for an n×n-matrix A with determinantal rank ρ(A)=t over an arbitrary commutative ring to be (von Neumann) strongly regular in terms of the trace of its tth compound matrix Ct(A). In particular, a non-zero n×n-matrix A with ρ(A)=t over a local commutative ring R is strongly regular if and only if Tr(Ct(A)) is a unit in R, and in this case we construct a strong inner inverse of A. We derive applications to products of local commutative rings and group algebras. Finally, we count strongly regular matrices over some finite rings of residue classes and group algebras.

Two questions on semi-clean group rings

It was proved in [4] that every group ring of a torsion abelian group over a commutative local ring is a semi-clean ring. It was asked in [4] whether every group ring of a torsion abelian group over a commutative clean ring is a semi-clean ring and whether every group ring of a torsion abelian group over a commutative semi-clean ring is a semi-clean ring. In this paper, we give a positive answer to question 1 and a negative answer to question 2.

Total Graphs Are Laplacian Integral

We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors. We also prove that the total graph of a finite commutative local ring with identity is super integral and give an example showing that this is not true for arbitrary rings.

Cohomological supports of tensor products of modules over commutative rings

This works concerns cohomological support varieties of modules over commutative local rings. The main result is that the support of a derived tensor product of a pair of differential graded modules over a Koszul complex is the join of the supports of the modules. This generalizes, and gives another proof of, a result of Dao and the third author dealing with Tor-independent modules over complete intersection rings. The result for Koszul complexes has a broader applicability, including to exterior algebras over local rings.

Von Neumann regular matrices revisited

We give a constructive sufficient condition for a matrix over a commutative ring to be von Neumann regular, and we show that it is also necessary over local rings. Specifically, we prove that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible -submatrix if and only if the determinantal rank and the McCoy rank of A coincide. We deduce consequences to (products of local) commutative rings, and we determine the number of von Neumann regular matrices over some finite rings of residue classes and group algebras.

Gorenstein Projective Coresolutions and Co-Tate Homology Functors

For a local commutative Gorenstein ring [Formula: see text], Enochs et al. in [Gorenstein projective resolvents, Comm. Algebra 44 (2016) 3989–4000] defined a functor [Formula: see text] and showed that this functor can be computed by taking a totally acyclic complex arising from a projective coresolution of the first component or a totally acyclic complex arising from a projective resolution of the second component. In order to define the functor [Formula: see text] over general rings, we introduce the right Gorenstein projective dimension of an [Formula: see text]-module [Formula: see text], [Formula: see text], via Gorenstein projective coresolutions, and give some equivalent characterizations for the finiteness of [Formula: see text]. Then over a general ring [Formula: see text] we define a co-Tate homology group [Formula: see text] for [Formula: see text]-modules [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], and prove that [Formula: see text] can be computed by complete projective coresolutions of the first variable or by complete projective resolutions of the second variable.

Trace Ideals and the Gorenstein Property

Let R be a local Noetherian commutative ring. We prove that R is an Artinian Gorenstein ring if and only if every ideal in R is a trace ideal. We discuss when the trace ideal of a module coincides with its double annihilator.

Factorization of some polynomials over finite local commutative rings and applications to certain self-dual and LCD codes

Factorization of some polynomials over finite local commutative rings and applications to certain self-dual and LCD codes

The Hilbert-Kunz function of some quadratic quotients of the Rees algebra

Given a commutative local ring ( R , m ) (R,\mathfrak m) and an ideal I I of R R , a family of quotients of the Rees algebra R [ I t ] R[It] has been recently studied as a unified approach to the Nagata’s idealization and the amalgamated duplication and as a way to construct interesting examples, especially integral domains. When R R is noetherian of prime characteristic, we compute the Hilbert-Kunz function of the members of this family and, provided that either I I is m \mathfrak {m} -primary or R R is regular and F-finite, we also find their Hilbert-Kunz multiplicity. Some consequences and examples are explored.