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A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add ⁡ ( M ) {\operatorname{Add}(M)} , which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add ⁡ ( M ) {\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V * ⁢ ( M ) {V^{*}(M)} , of isomorphism classes of countably generated modules in Add ⁡ ( M ) {\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V * ⁢ ( M ) {V^{*}(M)} is a submonoid of V * ⁢ ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} , this allows us to make computations with examples and to prove some realization results. [ABSTRACT FROM AUTHOR]

This paper presents a method of switching from position to stiffness control for a two-degree-of-freedom robot arm driven by three pairs of antagonistic actuators by applying the position and stiffness controller with the aim of developing a robot arm that can perform tasks involving a change from free to constrained motion. The control law consists of a feedback position controller and a feedforward stiffness controller in joint space. The position controller is a traditional PD feedback. The stiffness controller is designed using a newly developed method for solving the robot redundancy and adjusts joint stiffness through the coactivation of all the actuators. The solution to the redundancy expresses the actuator force space by a direct sum of its three subspaces and gives two independent relationships between actuator forces to joint torques and to joint stiffness. The position and stiffness controller is expected to control the position and the stiffness simultaneously and seamlessly perform tasks that involve changing from free to constrained motion. However, it has been revealed that the PD feedback controller affects the joint stiffness and that the stiffness controller reduces energy efficiency during position control. Therefore, this paper attempts to solve the problem by controlling the free motion with the position controller and by controlling the stiffness after the robot reaches the desired posture with the stiffness controller. Simulation results demonstrate that the proposed method can bring the robot to the desired posture, switch the control smoothly, and generate the desired stiffness.

Let K be a field, Q = (Q0, Q1) be a quiver and KQ be the generalised path algebra [10]. This paper gives a characterisation for the right and left modules over the path algebras of finite acyclic quiver. The study shows that the modules over such path algebras could be written as the decomposition of KQ-submodules. For KQ-modules over path algebras of quiver with countably many vertices, a sequence of KQ-submodules is identified which in finite case is a composition series. [ABSTRACT FROM AUTHOR]

Recall that an R-module M is pure-semisimple if every module in the category σ [ M ] is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that "a commutative ring R is pure-semisimple (i.e., every R-module is a direct sum of finitely generated modules) if and only if every R-module is a direct sum of cyclic modules, if and only if, R is an Artinian principal ideal ring". Consequently, every (or finitely generated, cyclic) ideal of R is pure-semisimple if and only if R is an Artinian principal ideal ring. Therefore, a natural question of this sort is "whether the same is true if one only assumes that every proper ideal of R is pure-semisimple?" The goal of this paper is to answer this question. The structure of such rings is completely described as Artinian principal ideal rings or local rings R with the maximal ideals M = R x ⊕ T which Rx is Artinian uniserial and T is semisimple. Also, we give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple. [ABSTRACT FROM AUTHOR]

The center algebra of a general multilinear form is defined and investigated. We show that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra, and center algebras can be effectively applied to direct sum decompositions of multilinear forms. As an application of the algebraic structure of centers, we show that almost all multilinear forms are absolutely indecomposable. The theory of centers can also be applied to symmetric equivalence of multilinear forms. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof for a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent. [ABSTRACT FROM AUTHOR]

Higher degree forms are homogeneous polynomials of degree d > 2, or equivalently symmetric d-linear spaces. This paper is mainly concerned about the algebraic structure of the centres of higher degree forms with applications specifically to direct sum decompositions, namely expressing higher degree forms as sums of forms in disjoint sets of variables. We show that the centre algebra of almost every form is the ground field, consequently almost all higher degree forms are absolutely indecomposable. If a higher degree form is decomposable, then we provide simple criteria and algorithms for direct sum decompositions by its centre algebra. It is shown that the direct sum decomposition problem can be boiled down to some standard tasks of linear algebra, in particular the computations of eigenvalues and eigenvectors. We also apply the structure results of centre algebras to provide a complete answer to the classical problem of whether a higher degree form can be reconstructed from its Jacobian ideal. [ABSTRACT FROM AUTHOR]

We focus on the Lp (ℝ2) theory of the fractional Fourier transform (FRFT) for 1 ⩽ p ⩽ 2. In L1 (ℝ2), we mainly study the properties of the FRFT via introducing the two-parameter chirp operator. In order to get the pointwise convergence for the inverse FRFT, we introduce the fractional convolution and establish the corresponding approximate identities. Then the well-defined inverse FRFT is given via approximation by suitable means, such as fractional Gauss means and Able means. Furthermore, if the signal ℱ α , β f is received, we give the process of recovering the original signal f with MATLAB. In L2 (ℝ2), the general Plancherel theorem, direct sum decomposition, and the general Heisenberg inequality for the FRFT are obtained. [ABSTRACT FROM AUTHOR]

A submodule W of V is summand absorbing, if x + y ∈ W implies x ∈ W , y ∈ W for any x , y ∈ V. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more generally of upper bound modules. [ABSTRACT FROM AUTHOR]

An R -module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0. More generally, a submodule W of V is "summand absorbing" (SA), if, for all x , y ∈ V , x + y ∈ W ⇒ x ∈ W , y ∈ W. These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of SA submodules of a given LZS module, especially, those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this paper, we consider which submodules are SA and describe their explicit generation. [ABSTRACT FROM AUTHOR]

We give two equivalent sets of invariants which classify pairs of coisotropic subspaces of finite-dimensional Poisson vector spaces. For this it is convenient to dualize; we work with pairs of isotropic subspaces of presymplectic vector spaces. We identify ten elementary types which are the building blocks of such pairs, and we write down a matrix, invertible over Z, which takes one 10-tuple of invariants to the other.

We give two equivalent sets of invariants which classify pairs of coisotropic subspaces of finite-dimensional Poisson vector spaces. For this it is convenient to dualize; we work with pairs of isotropic subspaces of presymplectic vector spaces. We identify ten elementary types which are the building blocks of such pairs, and we write down a matrix, invertible over Z, which takes one 10-tuple of invariants to the other.

• Based on the relationship between different dimensions of semantic subspace, a concise learning model is constructed. • A simple yet effective approach is proposed to simplify the complex and intractable semantic features. • Two new regular terms are introduced into the loss function to ensure the direct sum decomposition. The core problem of zero-shot learning (ZSL) is extracting concise feature from cumbersome and intricate semantic space. While semantic vectors were mainly used to constrain the distance between different classes in embedding space in existing ZSL methods, we first emphasize the importance of the relationship between different dimensions in semantic space. In this paper, we propose a novel inductive approach, Multiple Semantic Subspaces Network (MSSN), to simplify the complex and intractable semantic features. Our method generates multiple disentangled subfeatures via direct sum decomposition, which not only retain completely semantic information, but also obtain simple independent hierarchical features. Specially, instead of original space, using projection subspace to map embedding space can reduce the difficulty of model optimization and enhance the generalization ability of the model. Full experiments are achieved on almost all datasets and comparision with many algorithms which exist in the present latest literature. Compared with nineteen competitors, the results show that our model outperforms the state-of-the-art on conventional ZSL setting and has a very competitive performance on generalized ZSL setting. [ABSTRACT FROM AUTHOR]

For a certain family of complete modular lattices, we prove a "Jordan–Hölder–Schreier-like" theorem with no assumptions on cardinality or well-orderedness. This family includes both lattices with are both join- and meet-continuous, as well as the lattices of subobjects of any object in an abelian category satisfying properties related to Grothendieck's axioms (AB5) and (AB5⁎). We then give several examples of objects in abelian categories which satisfy these axioms, including pointwise finite-dimensional persistence modules, presheaves, and certain Prüfer modules. Moreover, we show that, over an arbitrary ring, the infinite product of isomorphic simple modules both fails to satisfy our axioms and admits at least two composition series with distinct cardinalities. We conclude by giving a lattice-theoretic proof that any object which is locally finitely generated and satisfies our axioms can be expressed as a direct sum of indecomposable subobjects. We conjecture that this decomposition is unique. [ABSTRACT FROM AUTHOR]

Abstract An R -module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0. More generally, we call a submodule W of V "summand absorbing" (SA) in V if ∀ x , y ∈ V : x + y ∈ W ⇒ x ∈ W , y ∈ W. These arise in tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. We explore the lattice of finite sums of SA-submodules, obtaining analogs of the Jordan–Hölder theorem, the noetherian theory, and the lattice-theoretic Krull dimension. We pay special attention to finitely generated SA-submodules, and describe their explicit generation. [ABSTRACT FROM AUTHOR]

In this article, we study the center of a generalized effect algebra (GEA), relate it to the exocenter, and in case the GEA is centrally orthocomplete (a COGEA), relate it to the exocentral cover system. Our main results are that the center of a COGEA is a complete boolean algebra and that a COGEA decomposes uniquely as the direct sum of an effect algebra (EA) that contains the center of the COGEA and a complementary direct summand in which no nonzero direct summand is an EA. [ABSTRACT FROM AUTHOR]

It is known that normal affine semigroup rings are of finite F-representation type. In this paper, we prove that non-normal affine semigroup rings are also of finite F-representation type. [ABSTRACT FROM PUBLISHER]

Given an arbitrary graph E we investigate the relationship between E and the groupoid GE. We show that there is a lattice isomorphism between the lattice of pairs (H, S), where H is a hereditary and saturated set of vertices and S is a set of breaking vertices associated to H, onto the lattice of open invariant subsets of G(0)E. We use this lattice isomorphism to characterise the decomposability of the Leavitt path algebra LK(E), where K is a field. First we find a graph condition to characterise when an open invariant subset of G(0)E. is closed. Then we give both a graph condition and a groupoid condition each of which is equivalent to LK(E) being decomposable in the sense that it can be written as a direct sum of two nonzero ideals. We end by relating decomposability of a Leavitt path algebra with the existence of nontrivial central idempotents. In fact, all the nontrivial central idempotents can be described. [ABSTRACT FROM AUTHOR]

In this article, we study the center of a generalized effect algebra (GEA), relate it to the exocenter, and in case the GEA is centrally orthocomplete (a COGEA), relate it to the exocentral cover system. Our main results are that the center of a COGEA is a complete boolean algebra and that a COGEA decomposes uniquely as the direct sum of an effect algebra (EA) that contains the center of the COGEA and a complementary direct summand in which no nonzero direct summand is an EA.

In this article, conservation rules of the direct sum decomposition of groups are mainly discussed. In the first section, we prepare miscellaneous definitions and theorems for further formalization in Mizar [5]. In the next three sections, we formalized the fact that the property of direct sum decomposition is preserved against the substitutions of the subscript set, flattening of direct sum, and layering of direct sum, respectively. We referred to [14], [13] [6] and [11] in the formalization. [ABSTRACT FROM AUTHOR]

In this article, the equivalent expressions of the direct sum decomposition of groups are mainly discussed. In the first section, we formalize the fact that the internal direct sum decomposition can be defined as normal subgroups and some of their properties. In the second section, we formalize an equivalent form of internal direct sum of commutative groups. In the last section, we formalize that the external direct sum leads an internal direct sum. We referred to [19], [18] [8] and [14] in the formalization. [ABSTRACT FROM AUTHOR]

In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization. [ABSTRACT FROM AUTHOR]

An Abelian group is pseudofree of rank ℓ if it belongs to the extended genus of ℤℓ, i.e., its localization at every prime p is isomorphic to ℤpℓ. A pseudofree group can be studied through a sequence of rational matrices, the so-called sequential representation. Here, we use these sequential representations to study the relation between the product of extended genera of free Abelian groups and the extended genus of their direct sum. In particular, using sequential representations, we give a new proof of a result by Baer, stating that two direct sum decompositions into rank one groups of a completely decomposable pseudofree Abelian group are necessarily equivalent. On the other hand, sequential representations can also be used to exhibit examples of pseudofree groups having nonequivalent direct sum decompositions into indecomposable groups. However, since this cannot occur when using the notion of near-isomorphism rather than isomorphism, we conclude our work by giving a characterization of near-isomorphism for pseudofree groups in terms of their sequential representations.

In this article, we show that certain generalized boolean subalgebras of the exocenter of a generalized effect algebra (GEA) determine hull systems on the GEA in a manner analogous to the determination of a hull mapping on an effect algebra (EA) by its set of invariant elements. We show that a hull system on a GEA E induces a hull mapping on each interval E[0, p] in E, and, using hull systems, we identify certain special elements of E (e.g., η-subcentral elements, η-monads, and η-dyads). We also extend the type-decomposition theory for EAs to GEAs. [ABSTRACT FROM AUTHOR]

Let R be an arbitrary commutative finite chain ring with $$1\ne 0$$. 1-generator quasi-cyclic (QC) codes over R are considered in this paper. Let $$\gamma $$ be a fixed generator of the maximal ideal of R, $$F=R/\langle \gamma \rangle $$ and $$|F|=q$$. For any positive integers m, n satisfying $$\mathrm{gcd}(q,n)=1$$, let $$\mathcal{R}_n=R[x]/\langle x^n-1\rangle $$. Then 1-generator QC codes over R of length mn and index m can be regarded as 1-generator $$\mathcal{R}_n$$-submodules of the module $$\mathcal{R}_n^m$$. First, we consider the parity check polynomial of a 1-generator QC code and the properties of the code determined by the parity check polynomial. Then we give the enumeration of 1-generator QC codes with a fixed parity check polynomial in standard form over R. Finally, under the condition that $$\mathrm{gcd}(|q|_n,m)=1$$, where $$|q|_n$$ denotes the order of q modulo n, we describe an algorithm to list all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial in standard form over R of length mn and index m. [ABSTRACT FROM AUTHOR]

Quaternary 1-generator quasi-cyclic codes are considered in the paper. Under the conditions that n is odd and gcd(|2|, m) = 1, where |2| denotes the order of 2 modulo n, we give the enumeration of quaternary 1-generator quasi-cyclic codes of length mn, and describe an algorithm which will obtain one, and only one, generator for each quaternary 1-generator quasi-cyclic code. [ABSTRACT FROM AUTHOR]

Let 풦 be the class of all right R-modules that are kernels of nonzero homomorphisms φ:E1→E2 for some pair of indecomposable injective right R-modules E1,E2. In a previous paper, we completely characterized when two direct sums A1⊕⋯⊕An and B1⊕⋯⊕Bm of finitely many modules Ai and Bj in 풦 are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many Ai and Bj in 풦. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class 풦 with the class 풰 of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered). [ABSTRACT FROM AUTHOR]

This paper discusses the enumeration of 1-generator quasi-cyclic codes and describes an algorithm which will obtain one, and only one, generator for each 1-generator quasi-cyclic code. [ABSTRACT FROM AUTHOR]

Abstract: We prove th root property for uniserial modules and we show that the Krull–Schmidt theorem holds for direct sums of uniserial modules that are not quasi-small. [Copyright &y& Elsevier]

If R = Fq[x&rceil/(xm - 1), S = Fqn[x⌉/(xm - 1), we define the mapping [This equation can not be represented in ASCII.TXT] from Rn onto S, where (α0,α1,. . . ,αn-1) is a basis for Fqn over Fq. This carries the q-ray 1-generator quasi-cyclic (QC) code R &asline;(x) onto the code RA(x) in S whose parity-cheek polynomial (p.c.p.) is defined as the monk polynomial h(x) over Fq of least degree such that h(x)A(x) = 0. In the special case, where gcd(q, m) = 1 and where the prime factorizations of xm-1 over Fq and Fqn are the same we show that there exists a one-to-one correspondence between the q-ary 1-generator quasis-cyclic codes with p.c.p. h(x) and the elements of the factor group J* /I* where J is the ideal In S with p.c.p. h(x) and I the corresponding quantity in R. We then describe an algorithm for generating the elements of J*/I*. Next, we show that if we choose a normal basis for Fqn over Fq, then we can modify the aforementioned algorithm to eliminate a certain number of equivalent codes, thereby rending the algorithm more attractive from a computational point of view. Finally in Section IV, we show how to modify the above algorithm in order to generate all the binary self-dual 1-generator QC codes. [ABSTRACT FROM AUTHOR]

It is shown that, if R is either an Artin algebra or a commutative noetherian domain of Krull dimension 1, then infinite direct products of R-modules resist direct sum decomposition as follows: If (M[SUBn])[SUBn∈N] is a family of non-isomorphic, finitely generated, indecomposable R-modules, then ∏[SUBn∈N] M[SUBn] is not a direct sum of finitely generated modules. The bearing of this direct product condition on the pure semisimplicity problem is discussed. [ABSTRACT FROM AUTHOR]