Featured Researches

Commutative Algebra

1-absorbing primary submodules

Let R be a commutative ring with non-zero identity and M be a unitary R -module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule N of M is said to be a 1-absorbing primary submodule if whenever non-unit elements a,b?�R and m?�M with abm?�N , then either ab??N : R M) or m?�M?�rad(N). Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.

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Commutative Algebra

A New Type of Bases for Zero-dimensional Ideals

We formulate a substantial improvement on Buchberger's algorithm for Gröbner bases of zero-dimensional ideals. The improvement scales down the phenomenon of intermediate expression swell as well as the complexity of Buchberger's algorithm to a significant degree. The idea is to compute a new type of bases over principal ideal rings instead of over fields like Gröbner bases. The generalizations of Buchberger's algorithm from over fields to over rings are abundant in the literature. However they are limited to either computations of strong Gröbner bases or modular computations of the numeral coefficients of ideal bases with no essential improvement on the algorithmic complexity. In this paper we make pseudo-divisions with multipliers to enhance computational efficiency. In particular, we develop a new methodology in determining the authenticity of the factors of the pseudo-eliminant, i.e., we compare the factors with the multipliers of the pseudo-divisions instead of the leading coefficients of the basis elements. In order to find out the exact form of the eliminant, we contrive a modular algorithm of proper divisions over principal quotient rings with zero divisors. The pseudo-eliminant and proper eliminants and their corresponding bases constitute a decomposition of the original ideal. In order to address the ideal membership problem, we elaborate on various characterizations of the new type of bases. In the complexity analysis we devise a scenario linking the rampant intermediate coefficient swell to Bézout coefficients, partially unveiling the mystery of hight-level complexity associated with the computation of Gröbner bases. Finally we make exemplary computations to demonstrate the conspicuous difference between Gröbner bases and the new type of bases.

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Commutative Algebra

A Note on Projective Modules

This expository note delves into the theory of projective modules parallel to the one developed for injective modules by Matlis. Given a perfect ring R , we present a characterization of indecomposable projective R -modules and describe a one-to-one correspondence between the projective indecomposable R -modules and the simple R -modules.

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Commutative Algebra

A Note on Residual Variables of an Affine Fibration

In a recent paper [El 13], M.E. Kahoui has shown that if R is a polynomial ring over C , A an A 3 -fibration over R , and W a residual variable of A then A is stably polynomial over R[W] . In this article we show that the above result holds over any Noetherian domain R provided the module of differentials Ω R (A) of the affine fibration A (which is necessarily a projective A -module by a theorem of Asanuma) is a stably free A -module.

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Commutative Algebra

A Structural Invariant On Certain Two-Dimensional Noetherian Partially Ordered Sets

If (X, ??X ) is a partially ordered set satisfying certain necessary conditions for X to be order-isomorphic to the spectrum of a Noetherian domain of dimension two, we describe a new poset (str X, ??str X ) that completely determines X up to isomorphism. The order relation ??str X imposed on str X is modeled after R. Wiegand's well-known "P5" condition that can be used to determine when a given partially ordered set (U, ??U ) of a certain type is order-isomorphic to (Spec Z[x],??.

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Commutative Algebra

A characterization of length-factorial Krull monoids

An atomic monoid is length-factorial if each two distinct factorizations of any element have distinct factorization lengths. We provide a characterization of length-factorial Krull monoids in terms of their class groups and the distribution of prime divisors in the classes.

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Commutative Algebra

A characterization of local nilpotence for dimension two polynomial derivations

Let K be an algebraically closed field. We prove that a polynomial K-derivation D in two variables is locally nilpotent if and only if the subgroup of polynomial K-automorphisms which commute with D admits elements whose degree is arbitrary big.

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Commutative Algebra

A characterization of weakly Krull monoid algebras

Let D be a domain and let S be a torsion-free monoid whose quotient group satisfies the ascending chain condition on cyclic subgroups. We give a characterization of when the monoid algebra D[S] is weakly Krull. As corollaries, we obtain the results on when D[S] is Krull resp. generalized Krull, due to Chouinard resp. El Baghdadi and Kim. Furthermore, we deduce Chang's theorem on weakly factorial monoid algebras and we characterize the weakly Krull domains among the affine monoid algebras.

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Commutative Algebra

A computing strategy and programs to resolve the Gerstenhaber Problem for commuting triples of matrices

We describe a MATLAB program that could produce a negative answer to the Gerstenhaber Problem by the construction of three commuting n×n matrices A,B,C over a field F such that the subalgebra F[A,B,C] they generate has dimension greater than n . This problem has remained open for nearly 60 years, following Gerstenhaber's surprising result (Annals Math.) that dimF[A,B]≤n for any two commuting matrices A,B . The property fails for four or more commuting matrices. We also make the MATLAB files freely available.

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Commutative Algebra

A constructive approach to one-dimensional Gorenstein k -algebras

Let R be the power series ring or the polynomial ring over a field k and let I be an ideal of R. Macaulay proved that the Artinian Gorenstein k -algebras R/I are in one-to-one correspondence with the cyclic R -submodules of the divided power series ring ?. The result is effective in the sense that any polynomial of degree s produces an Artinian Gorenstein k -algebra of socle degree s. In a recent paper, the authors extended Macaulay's correspondence characterizing the R -submodules of ? in one-to-one correspondence with Gorenstein d-dimensional k -algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein k -algebras of dimension one and any codimension. This has been achieved through a deep analysis of the G -admissible submodules of ?. Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed.

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