Projective spread sets

In this paper the notion of a spread set for at-spread ofPG(2t+1,q) is generalised and it is shown that certaint-spreads ofPG(n, q) correspond to these generalised spread sets. Then a projective spread set is defined and it is shown that anyt-spread ofPG(n, q) corresponds to a projective spread set. Connections between the spread set and the projective spread set of at-spread are discussed, in particular in the case of at-spread ofPG(2t + 1,q) the spread set and the projective spread set are equivalent, giving a new and straightforward construction of a spread set. The methods developed are used to show, with the aid of a computer, that the 1-packing ofPG(7,2) constructed by Baker is regulus-free.Dedicated to Professor Giuseppe Tallini on the occasion of his 60th birthday     

An Upper Bound for the Regularity of Ideals of Borel Type

We show that the regularity of monomial ideals of K[x ₁,…, x _n ] (K being a field), whose associated prime ideals are totally ordered by inclusion is upper bounded by a linear function in n.     

Algebraic Properties of the Path Ideal of a Tree

The path ideal (of length t ≥ 2) of a directed graph Γ is the monomial ideal, denoted I _t (Γ), whose generators correspond to the directed paths of length t in Γ. We study some of the algebraic properties of I _t (Γ) when Γ is a tree. We first show that I _t (Γ) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I _t (Γ) is always sequentially Cohen–Macaulay, and the Betti numbers of R/I _t (Γ) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the article. We give an exact formula for the projective dimension of these ideals, and in some cases, we compute their arithmetical rank.     

Specializations of Ferrers ideals

We introduce a specialization technique in order to study monomial ideals that are generated in degree two by using our earlier results about Ferrers ideals. It allows us to describe explicitly a cellular minimal free resolution of various ideals including any strongly stable and any squarefree strongly stable ideal whose minimal generators have degree two. In particular, this shows that threshold graphs can be obtained as specializations of Ferrers graphs, which explains their similar properties.     

Path ideals of rooted trees and their graded Betti numbers

Let Γ be a rooted (and directed) tree, and let t be a positive integer. The path ideal It(Γ) is generated by monomials that correspond to directed paths of length (t−1) in Γ. In this paper, we study algebraic properties and invariants of It(Γ). We give a recursive formula to compute the graded Betti numbers of It(Γ) in terms of path ideals of subtrees. We also give a general bound for the regularity, explicitly compute the linear strand, and investigate when It(Γ) has a linear resolution.     

Asymptotic Growth of Associated Primes of Certain Graph Ideals

We specify a class of graphs, H _t , and characterize the irreducible decompositions of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question “For each integer t ≥ 0, does there exist a (hyper) graph H _t such that stabilization of associated primes occurs at n ≥ (χ(H _t ) ?1) + t?” [4 Francisco , C. A. , Hà , H. T. , Van Tuyl , A. ( 2011 ). Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals . J. Algebra 331 : 224 – 242 .[Crossref], [Web of Science ®] , [Google Scholar]]. For each t, H _t has chromatic number 3 and associated primes that stabilize at n = 2 + t.     

The Lefschetz Property for Componentwise Linear Ideals and Gotzmann Ideals

Abstract

For standard graded Artinian K-algebras defined by componentwise linear ideals and Gotzmann ideals, we give conditions for the weak Lefschetz property in terms of numerical invariants of the defining ideals.     

Primary Ideals and Valuation Ideals in Semigroups

Let S be a cancellation torsion-free additive semigroup with identity 0 and let S {0}. We consider the relation between the valuation ideals and primary ideals of S. We give a characterization that each primary ideal is a valuation ideal in S and also give a characterization that each valuation ideal is a primary ideal in S.AMS Subject Classification (1991): 20M14     

2-Spreads and Transitive and Orthogonal 2-Parallelisms of PG(5, 2)

A 2-spread is a set of two-dimensional subspaces of PG(d, q), which partition the point set. We establish that up to equivalence there exists only one 2-spread of PG(5, 2). The order of the automorphism group preserving it is 10584. A 2-parallelism is a partition of the set of two-dimensional subspaces by 2-spreads. There is a one-to-one correspondence between the 2-parallelisms of PG(5, 2) and the resolutions of the 2-(63,7,15) design of the points and two-dimensional subspaces. Sarmiento (Graphs and Combinatorics 18(3):621–632, 2002) has classified 2-parallelisms of PG(5, 2), which are invariant under a point transitive cyclic group of order 63. We classify 2-parallelisms with automorphisms of order 31. Among them there are 92 2-parallelisms with full automorphism group of order 155, which is transitive on their 2-spreads. Johnson and Montinaro (Results Math 52(1–2):75–89, 2008) point out that no transitive t-parallelisms of PG(d, q) have been constructed for t > 1. The 92 transitive 2-parallelisms of PG(5, 2) are then the first known examples. We also check them for mutual orthogonality and present a set of ten mutually orthogonal resolutions of the geometric 2-(63,7,15) design.     

Prime Filtrations and Primary Decompositions of Modules

We give a new characterization for prime filtrations of an R-module M in terms of primary decomposition of the zero submodule of M. Using this characterization we prove that some classes of monomial ideals like generic and cogeneric monomial ideals are clean, pretty clean, or almost clean.     

Skeletons of monomial ideals

In analogy to the skeletons of a simplicial complex and their Stanley–Reisner ideals we introduce the skeletons of an arbitrary monomial ideal I ? S = K [x₁, …, x_n ]. This allows us to compute the depth of S /I in terms of its skeleton ideals. We apply these techniques to show that Stanley's conjecture on Stanley decompositions of S /I holds provided it holds whenever S /I is Cohen–Macaulay. We also discuss a conjecture of Soleyman Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strongly stable rank and applications to matrix completion

We perform an in-depth study of strongly stable ranks of modules over a commutative ring. Here we define the strongly stable rank of a module to be the supremum of the stable ranks of its finitely generated submodules. As an application, we give non-Noetherian generalizations of known facts about outer products and matrix completions over PIRs and Dedekind domains. We construct Noetherian and non-Noetherian domains of arbitrary strongly stable rank. We also consider strongly n-generated ideals, and we characterize the rings in which every ideal is strongly 2-generated and the domains in which every ideal is strongly 3-generated.     

On the index of powers of edge ideals

The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper, we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal I is strictly increasing if I is linearly presented. Examples show that this needs not to be the case for monomial ideals generated in degree greater than two.     

The Stanley Conjecture on Intersections of Four Monomial Prime Ideals

We show that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra S over a field and for an arbitrary intersection of monomial prime ideals (P _i ) _i∈[s] of S such that each P _i is not contained in the sum of the other (P _j ) _j≠i .     

Diagonal multilinear operators on Köthe sequence spaces

We analyse the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of (E, p)-summing multilinear mappings, a natural extension of the linear ideal of absolutely (E, p)-summing operators.     

On the transposed Rao’s flag-transitive plane of order 49

Rao’s flag-transitive plane π of order 49 and π ^t , the plane obtained by transposing matrices of a representative set of π, has been studied. It is shown that π ^t is flag-transitive, π ^t is not isomorphic to π, and π ^t is obtained from π by replacement of a net of degree 25. Further, (1) the flag-transitive planes associated with 1-spread sets S _2b and S _2a in the classified list of translation planes of order 49 enumerated by Mathon et al, are respectively isomorphic to π and π ^t (2) The flag-transitive planes associated with the 1-spread sets of 0an* in the classified list of translation planes of order 49 enumerated by Charnes et al are isomorphic to π and π ^t in some order.     

On right S-Noetherian rings and S-Noetherian modules

In this paper we study right S-Noetherian rings and modules, extending notions introduced by Anderson and Dumitrescu in commutative algebra to noncommutative rings. Two characterizations of right S-Noetherian rings are given in terms of completely prime right ideals and point annihilator sets. We also prove an existence result for completely prime point annihilators of certain S-Noetherian modules with the following consequence in commutative algebra: If a module M over a commutative ring is S-Noetherian with respect to a multiplicative set S that contains no zero-divisors for M, then M has an associated prime.     

On Primitive Ideals in Polynomial Rings over Nil Rings

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R. Presented by S. MontgomeryMathematics Subject Classifications (2000) 16N40, 16N20, 16N60, 16D25.Agata Smoktunowicz: Current address: Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw 10, niadeckich 8, P.O. Box 21, Poland.