Duplication of a module along an ideal

Let A be a commutative ring and I an ideal of A. The amalgamated duplication of A along I, denoted by \({A \bowtie I}\) , is the special subring of \({A \times A}\) defined by \({A \bowtie I } := \pi \times_{\frac{A}{I}} \pi = \{(a, a + i) \mid a \in A, i \in I\}\) . We are interested in some basic and homological properties of a special kind of \({A \bowtie I}\) -modules, called the duplication of M along I with M is an A-module, and defined by \({M \bowtie I := \{(m, m') \in M \times M \mid m - m^{\prime} \in IM\}}\) . The new results generalize some results on amalgamated duplication of a ring along an ideal.     

The f-depth of an ideal on a module

Let be an ideal of a Noetherian local ring and a finitely generated -module. The f-depth of on is the least integer such that the local cohomology module is not Artinian. This paper presents some part of the theory of f-depth including characterizations of f-depth and a relation between f-depth and f-modules.

     

Implementation of sparse matrix algorithms in an advection-diffusion-chemistry module

A two-dimensional advection-diffusion-chemistry module of a large-scale environmental model is taken. The module is described mathematically by a system of partial differential equations. Sequential splitting is used in the numerical treatment. The non-linear chemistry is most time-consuming part and it is handled by six implicit algorithms for solving ordinary differential equations. This leads to the solution of very long sequences of systems of linear algebraic equations. It is crucial to solve these systems efficiently. This is achieved by applying four different algorithms. The numerical results indicate that the algorithms based on a preconditioned sparse matrix technique and on a specially designed algorithm for the particular problem under consideration perform best.     

极小内射模与极小平坦模

在本文中,我们引进了极小内射模、极小平模模以及M.P环的概念,给出了它们的一些特征刻画,并用这两类模刻画了D edek ind环,VN正则环.     

Sequencing jobs for an automated manufacturing module with buffer

This article deals with the real-time scheduling of jobs for an automated manufacturing module offering a variety of operations with negligible setup times. Since the jobs are brought to the system by a paced line at a fixed rate, each job is expected to be processed within a fixed cycle time. Hence, a group of jobs with longer processing-time may cause immediate accumulation of jobs in the limited input buffer. Consequently, an arriving job may be blocked from entering until the job in the station is completed. A Dynamic Program is implemented to schedule the jobs with the minimum total blocking time. Depending on the data, this implementation may become computationaly intractable, and thereby, a heuristic is developed to produce good solutions in polynomial time. The worst case error bounds are established for the heuristic solutions. An extensive experimentation shows that the heuristic yields close to optimal solutions effectively.     

On module cohomology of the Fourier algebra of an inverse semigroup

Semigroup Forum - For an inverse semigroup S with the set of idempotents E, we find necessary and sufficient conditions for the Fourier algebra A(S) to be module amenable, module character...     

The ring of integers of an Abelian extension of an algebraic number field as a Galois module

The ringO of integers of a finite Abelian extension K of an algebraic number field k is studied as a module over the group ring =[G], where is the ring of integers of k and G is the Galois group of K/k. It is proved that the ring is a decomposable -module if and only if there exists in K/k an intermediate extension K/F. FK, whose degree divides the different.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta. im. V. A. Steklova AN SSSR, Vol. 71, pp. 80–84, 1977.     

Existence of a multiplicative basis for a finitely spaced module over an aggregate

It is proved that a finitely spaced module over ak-category admits a multiplicative basis (such a module gives rise to a matrix problem in which the allowed column transformations are determined by a module structure, the row transformations are arbitrary, and the number of canonical matrices is finite).Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 567–579, May, 1994.     

Molnár-dependence in a pre-Hilbert module over an H*-algebra

Molnár-dependence is related to the strong Cauchy-Schwarz inequality in a pre-Hilbert module over an H*-algebra analogously as the linear dependence is related to the Cauchy–Schwarz inequality in a pre-Hilbert space. Necessary and sufficient conditions for two elements of a pre-Hilbert module to be Molnár-dependent are established in this article, what is enabled by proving a stronger inequality than the strong Cauchy–Schwarz one. Furthermore, it is shown that Molnár-dependence is a transitive relation.     

The Structure of an SL2‐module of finite Morley rank

We consider a universe of finite Morley rank and the following definable objects: a field , a non‐trivial action of a group on a connected abelian group V , and a torus T of G such that . We prove that every T‐minimal subgroup of V has Morley rank . Moreover V is a direct sum of ‐minimal subgroups of the form , where W is T‐minimal and ζ is an element of G of order 4 inverting T .     

When an infinite direct product of modules is a free module of finite rank

Let R be a commutative ring and let α be an infinite cardinal number. If a direct product of α nonzero R-modules is a free R-module of finite rank, then R is a ring-direct product of α nonzero rings.     

Rank-preserving module maps

In this paper, we characterize rank one preserving module maps on a Hilbert C^*-module and study its applications on free probability theory.     

A non-recursive criterion for weights of a highest-weight module for an affine Lie algebra

Let Λ be a dominant integral weight of level k for the affine Lie algebra g and let α be a non-negative integral combination of simple roots. We address the question of whether the weight η = Λ ? α lies in the set P(Λ) of weights in the irreducible highest-weight module with highest weight Λ. We give a non-recursive criterion in terms of the coefficients of α modulo an integral lattice kM, where M is the lattice parameterizing the abelian normal subgroup T of the Weyl group. The criterion requires the preliminary computation of a set no larger than the fundamental region for kM, and we show how this set can be efficiently calculated.     

The Lutz filtration as a Galois module in an extension without higher ramification

One considers the structure of the group of the points of a formal group and its Lutz filtration as a Galois module in an extension without higher ramification of a local field. Making use, on one hand, of Honda's theory on the classification of formal groups over complete local rings and, on the other hand, of a generalization to formal groups of the Artin-Hasse function, one constructs effectively an isomorphism between the group of points and some given additive free Galois module. In particular, in the multiplicative case one gives a new effective proof of Krasner's theorem on the normal basis of the group of principal units of a local field in extensions without higher ramification. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 182–192, 1987.